Heat-content maximization on compact metric graphs at every time
Henry Zweiman
September 16, 2026
Abstract
Among finite compact connected metric graphs of prescribed total length with a nonempty Dirichlet vertex set and standard conditions elsewhere, the interval with one Dirichlet and one Neumann endpoint maximizes heat content at every positive time. Equality at a single positive time characterizes this interval, up to subdivision. This answers the all-times question of Bifulco and Täufer. We prove the stronger statement that heat evolution on any such graph is dominated in cumulative decreasing rearrangement by mixed-boundary interval evolution of the rearranged initial datum. The argument combines the classical metric-graph level-set inequality with the zero-order elliptic concentration comparison and implicit-time iteration of parabolic symmetrization. The rigidity proof uses the half-time energy identity and strict edgewise concavity of the heat evolution of the constant function. No arithmetic condition on the edge lengths is imposed. We also prove that, among graphs of fixed length carrying a finite nonnegative killing measure of fixed positive mass, the interval with all killing at one endpoint maximizes heat content at every time. This comparison allows singular measures and arbitrary nonnegative initial data, and equality at a single positive time characterizes the interval and its endpoint measure. For the same general killing measures we prove sharp Kohler-Jobin inequalities at fixed length and total killing, and at fixed torsion and total killing. A ground-state distribution estimate controls the potential terms and compares the resulting scalar differential inequality with the cosine profile; both equality cases characterize the endpoint-killed interval. The stationary joint comparisons also extend to the nonlinear -Laplacian for every , with explicit finite-strength constants and the same rigidity.
Let be a finite compact connected metric graph, with positive edge lengths and total length . Let be a nonempty subset of its vertices. We assume that is connected. The Laplacian has Dirichlet conditions at and standard continuity and Kirchhoff conditions at every other vertex. A standard condition at a vertex of degree one is a Neumann condition. Loops and multiple edges are permitted; degrees and Kirchhoff sums count edge incidences, so a loop contributes twice.
Write for the nonnegative realization of and . Its heat content is
We compare to , with Neumann condition at and Dirichlet condition at . Denote the corresponding operator, semigroup and heat content by , and .
Theorem 1.1 (heat-content maximization and rigidity). For every graph described above and every ,
Equality for one positive time holds if and only if is a path of length , consists of one endpoint, and all other vertices have standard conditions. Equivalently, after suppressing vertices of degree two, the graph is the Dirichlet-Neumann interval. In particular, for every other admissible graph the inequality is strict at every positive time.
For a nonnegative measurable function on a space of length , let be its decreasing rearrangement on , and set
We write if for all . Equality of total integrals is not part of this definition. Thus the order is weak submajorization, also called concentration order in the parabolic comparison literature.
Theorem 1.2 (concentration comparison). Let and let be nonincreasing. If
then, for every ,
The interval function on the right is nonnegative and nonincreasing. In particular, one may take .
Taking and proves the inequality in Theorem 1.1. Its one-time equality assertion needs an additional argument and is proved in Section 5.
1.1. Relation to earlier work
Bifulco and Täufer [BT, Theorem 2.6] prove the interval comparison for sufficiently large times, and for sufficiently small times when the edge lengths are rationally dependent. Their final Remark 2.9 leaves the comparison at every positive time open. Theorem 1.1 resolves that question without a restriction on edge lengths or on the time. The graph, vertex conditions and total-length normalization here agree with their Definitions 2.1 and 2.2. Their comparison interval is written with its Dirichlet endpoint at ; reflection gives our convention.
Bifulco and Mugnolo [BM] derive a heat-content formula, short-time asymptotics and surgery principles for compact quantum graphs. Their Section 6 includes comparisons under imposing Dirichlet conditions, taking subgraphs and attaching pendant graphs, as well as heat-content-preserving constructions. These are a different set of comparisons from (5), which rearranges arbitrary initial data and compares graphs of the same total length. We use the ordinary Kirchhoff heat semigroup studied in that work.
The geometric rearrangement input is classical. In the proof of [F, Lemma 3], Friedlander uses the number of points in a regular level set to compare the graph Dirichlet energy to the energy of a monotone rearrangement on an interval. That proof also explains how equality in level multiplicity excludes branching. We give the relevant calculation because the zero-order term of the resolvent equation must be retained.
The passage from elliptic concentration comparison to parabolic comparison by implicit-time discretization is likewise classical. Vázquez [V, Sections 5 and 7] develops precisely this strategy for Euclidean equations; see in particular the zero-order comparison and the parabolic iteration in the author's available text. We adapt that mechanism to the mixed-boundary graph problem. Neither the level-set Cauchy-Schwarz inequality nor implicit Euler iteration is asserted to be new. The contribution is the all-times graph comparison (5), its answer to [BT, Remark 2.9], and the one-time rigidity theorem. The elementary consequences in Section 6 are included as applications of the same result. Section 7 extends the concentration theorem to nonnegative killing measures of prescribed total mass, using a Robin interval comparator and a boundary inequality for cumulative resolvents. It includes delta-vertex conditions, nonnegative potentials and finite singular measures.
2. Graph forms and rearrangement preliminaries
The measure on is the sum of Lebesgue measures on the edges; vertices have measure zero. The closed quadratic form of is
Here consists of edgewise functions continuous at the vertices. The operator domain consists of edgewise functions in the form domain satisfying the Kirchhoff sum at each vertex outside . The same construction on , with form domain , gives .
Both operators are nonnegative and self-adjoint, with compact resolvent. They are strictly positive: if , integration along a simple path from to any point gives
The truncations and preserve the form domain and do not increase its energy. The resolvents and semigroups are therefore positivity preserving and sub-Markovian. In particular,
We recall two elementary rearrangement facts. For nonnegative on a nonatomic space of length ,
The supremum formula follows by filling a superlevel set of and, if necessary, part of one level set. The second assertion follows by comparing the integrals on each candidate set. Also,
for nonnegative . Indeed, Hardy-Littlewood gives , while rearrangement preserves the two squared norms. Expanding the squared distances proves (10).
Lemma 2.1 (a Lipschitz rearrangement). Suppose is continuous on , vanishes at , and is Lipschitz on every edge, with maximum edge Lipschitz constant . Then has a Lipschitz representative on with Lipschitz constant at most , and .
Proof. The case is immediate. Choose a maximum point and a simple path from it to . For , this path contains a subarc along which passes from to while remaining between those values. The length of that subarc is at least . The path is simple, so its length is counted without multiplicity in the graph measure. Consequently
This inverse-distribution estimate gives the Lipschitz assertion; jumps of the distribution due to plateaus correspond to constant portions of the rearrangement and cause no difficulty. Finally, continuity at a Dirichlet vertex shows that the essential infimum is zero. Hence the representative has .
3. Resolvent concentration comparison
For , put
Lemma 3.1 (the interval resolvent). If is nonincreasing and belongs to , then is nonnegative and nonincreasing. With and ,
If are two such functions and for all , then for all .
Proof. First let be smooth on . The equation is , with and . Positivity of the resolvent gives , hence . Its derivative satisfies
Testing against gives . Every nonnegative nonincreasing function can be approximated in by smooth nonnegative nonincreasing functions: first truncate, then extend by constant endpoint values and mollify. Boundedness of on passes the conclusion to the limit; the cone of nonincreasing functions is closed in .
Integration of the resolvent equation from to , using , proves (13). This remains valid for data because . For the last assertion, the difference satisfies , and . Thus
and .
Lemma 3.2 (graph resolvent comparison). For every ,
Consequently, if satisfies the hypotheses and concentration bound of Theorem 1.2, then
Proof. We first suppose that on every closed edge the load is a strictly positive nonconstant polynomial. Loads need not be continuous across vertices. The solution satisfies
on each edge, so it extends analytically past both endpoints as an edgewise function. It is strictly positive outside . For completeness, a zero interior minimum contradicts (18) and . At a non-Dirichlet vertex with value zero, all outgoing derivatives would be nonnegative, and their Kirchhoff sum would make them all zero. Equation (18) would then give a negative second derivative into each incident edge at that endpoint, contradicting nonnegativity. This proves the positivity assertion.
The function cannot be constant on an edge, because (18) would force to be constant there. Each edge therefore has only finitely many critical points: its derivative is analytic on a neighborhood of the closed edge and is not identically zero. Exclude their values and all vertex values. For every remaining , write
All sums run over interior edge points. Continuity along a path from a maximum to gives . Define and . Integration of (18) over gives
To check the sign and vertex terms, the derivative pointing out of a superlevel interval is at a regular boundary point. At a vertex contained in , continuity puts all incident edge germs in , and their fluxes cancel by Kirchhoff. No Dirichlet vertex belongs to . These observations give exactly the positive flux term on the left of (20).
The one-dimensional coarea formula and inverse differentiation give
Cauchy-Schwarz, in the form already used for graph rearrangement in [F], yields
Combining (20)-(22), we obtain
There is no omitted singular part in (23). The function is Lipschitz by Lemma 2.1, so . There are no plateaus of , and hence the finitely many exceptional levels yield only finitely many exceptional ranks . Thus the displayed inequality holds almost everywhere and in the weak sense. Its boundary data are
By Lemma 3.1, the cumulative integral of satisfies the equality in (23), with the same boundary data. The positive-part test (15) proves .
To remove the polynomial assumption, approximate an arbitrary nonnegative load by strictly positive nonconstant polynomials on every closed edge. Such loads are dense: separately on each edge, approximate first by a nonnegative continuous function, then uniformly by a polynomial, adding a vanishing positive constant to ensure strict positivity. A vanishing linear perturbation makes a constant polynomial nonconstant while preserving positivity. There are only finitely many edges.
For an approximating sequence in , resolvent boundedness and (10) give
Finite total length converts these to convergences, and (9) passes the cumulative inequality to the limit. This proves (16). Finally apply Lemma 3.1 to the decreasing interval data and to obtain (17).
The comparison retains the zero-order term in (18). Dropping that term would give a Poisson comparison, which does not by itself iterate to the required heat inequality.
4. Passage to the heat semigroup
Proof of Theorem 1.2. Fix and define
All these functions are nonnegative, and every is nonincreasing by Lemma 3.1. The hypothesis at and (17) give inductively
For a fixed , take and . The spectral theorem gives
Indeed, for every , and both multipliers are bounded by one. Dominated convergence in the spectral measure proves (28). Passing to the limit in (27) by (9) proves (5), uniformly in . Closedness of the cone of nonincreasing functions gives the remaining monotonicity assertion.
This is the classical implicit-time strategy of [V], with the graph resolvent inequality supplied by Section 3. In particular, no differentiability of a time-dependent rearrangement and no assumption about regular heat levels are needed to prove the inequality.
5. Equality at a single time
The proof of rigidity uses the constant initial datum. Put
We first record a strict regularity property of this particular heat evolution.
Lemma 5.1 (strict edgewise concavity). For every , the function is smooth on the closure of each individual edge, positive outside , and
Consequently, each open edge has at most one critical point of and has no constant subinterval.
Proof. Spectral smoothing gives for every integer . Repeated use of the edge equation for the operator gives edgewise regularity, and hence smoothness up to each endpoint. On a graph connected outside its Dirichlet set, the Kirchhoff heat semigroup is positivity improving on .
Here is a maximum-principle justification of the positivity statement used in this proof. For nonzero nonnegative initial data, positivity preservation and injectivity of give a nonzero nonnegative profile at positive times. An interior zero at a positive time forces the solution to vanish backwards on that edge by the parabolic strong maximum principle. At a standard vertex where the nonnegative solution has value zero, all outgoing derivatives are nonnegative. Their sum is zero, so every one vanishes; the parabolic boundary point lemma excludes a positive incident edge with zero outgoing derivative. Thus positivity propagates across each standard vertex. Connectivity of gives positivity throughout the open edges and at its standard vertices. This is the usual strict-positivity property of the connected graph heat kernel; see also the discussion in [BM, Section 1].
Sub-Markovianity and positivity give
Differentiation at positive time in implies . It is not the zero function: otherwise (7) would give , which contradicts positivity. Commutation through the semigroup yields
Positivity improvement applied to the nonzero nonnegative datum on the right proves (30). The final assertion follows because is strictly decreasing.
Lemma 5.2 (equal norms under concentration). Suppose belong to on spaces of the same finite measure, , and . Then almost everywhere.
Proof. For define
The same formula holds for , so . Tonelli gives
Equality of the norms makes almost everywhere, and then everywhere by continuity. Their right derivatives are and , respectively. Thus the distributions agree and so do the rearrangements.
Proof of the rigidity assertion in Theorem 1.1. Self-adjointness and the semigroup law give the half-time identities
The identical formulas hold on the interval. Suppose for some . Theorem 1.2 at time gives . Lemma 5.2 and (35) imply
The differentiable nonnegative function attains its minimum zero at the interior point , so its derivative vanishes there. Thus (35) gives
Set and . By Lemma 5.1, only finitely many critical or vertex values need be excluded. At all other levels use the notation from (19). Lemma 2.1 and one-dimensional coarea yield
The second formula follows from , and on every interval between exceptional values. The Lipschitz representative ensures that these identities lose no singular energy. By (22),
Equations (37)-(39) force for almost every regular .
Every edge incidence at a Dirichlet vertex has strictly positive derivative into its edge. Indeed, the function is positive inside that edge, zero at its endpoint, and strictly concave; the endpoint derivative is at least any positive chord slope. Therefore, for all sufficiently small positive regular levels,
The small incident edge neighborhoods can be chosen disjoint, including the two incidences of a loop. It follows that consists of one vertex and that this vertex has degree one.
Next consider a standard vertex of degree . Write and let be the derivatives along its outgoing incidences. Kirchhoff gives . Each incidence with gives increasing values immediately away from ; each incidence with gives decreasing values. If , strict edgewise concavity also gives decreasing values. Thus every incidence is assigned to one of two classes. Since , one class contains at least two incidences. On the corresponding side of , all sufficiently close regular levels have at least two distinct preimages. This contradicts almost everywhere. No standard vertex can have degree at least three.
A connected finite graph whose degrees are at most two and which has a vertex of degree one is a path. Its unique Dirichlet vertex is an endpoint; its other endpoint is standard and hence Neumann. Suppressing standard degree-two vertices changes neither the metric space nor its Laplacian. This proves necessity. Sufficiency follows by reflection and subdivision invariance of the interval problem.
The equality argument does not infer strictness from strict resolvent inequalities followed by a limit. Such a limit could erase a positive defect. Instead, equality at the specified time produces the exact energy equality (37).
6. Consequences and limits
Corollary 6.1 (convex temperature functionals). Under the hypotheses of Theorem 1.2, for every convex nondecreasing function with ,
In particular, all norms, , are bounded by those of the comparison evolution.
Proof. Concentration order implies the stop-loss comparison (33) and the comparison of first moments. Every such convex is a nonnegative linear function plus an integral of against its distributional second derivative; approximation applies if necessary. Integrating that representation proves (41). Choosing gives finite ; passage to the limit gives . At positive time the profiles are bounded by spectral smoothing and the edge Sobolev embedding.
The interval bound has the explicit expansion
Indeed, the normalized eigenfunctions are . Their squared integrals are , giving (42) by the spectral representation of (1).
Corollary 6.2 (positive time weights). Let be measurable. Whenever the interval integral is finite,
If is positive on a set of positive measure, equality characterizes the same interval. In particular, choosing recovers the sharp torsional-rigidity bound .
Proof. Integrate Theorem 1.1 and use its strictness. For the last constant, the interval torsion function solves , , hence and .
The torsional comparison itself is prior work; it is discussed in [BT, Remark 2.3] and is not a separate novelty claim here. Similarly, with the diffusion convention that the generator is on each edge, is the survival probability at time for the killed graph diffusion started from uniform length measure. Theorem 1.1 therefore also says that its lifetime is stochastically dominated by the lifetime on the mixed interval. This is the direct probabilistic interpretation of the heat-content bound, using the Feynman-Kac representation in [BT, Proposition 3.2].
The conclusions of Sections 1-6 concern unweighted finite metric graphs with the stated Dirichlet and Kirchhoff conditions. Section 7 gives a separate theorem for nonnegative killing measures with continuous form domain. They do not provide a heat-trace bound, pointwise heat-kernel domination, or a theorem for arbitrary vertex couplings. We prove no quantitative stability estimate. A useful next problem is to quantify the loss in (2) in terms of geometric distance from a path, with explicit control of the time scale and of edges whose lengths tend to zero. Such a statement requires additional work beyond the qualitative rigidity proved here.
7. A fixed amount of nonnegative killing
The comparison extends to a different boundary regime: there are no Dirichlet conditions, and killing is described by a finite nonnegative measure. Throughout this section, is a finite compact connected metric graph of total length , and is a nonnegative Radon measure on with
The measure is distinct from the length measure used for heat content and rearrangement. In particular, may charge vertices. On the space of continuous edgewise functions, define
Here is the operator associated with the form. This includes nonnegative integrable potentials and nonnegative delta interactions, separately or together, as well as singular continuous killing measures. It does not include signed potentials or weighted continuity conditions.
The comparison form on is
Its operator has and . Write and for its semigroup and heat content.
Theorem 7.1 (fixed-total-killing comparison and rigidity). Suppose satisfies (44), , and is nonincreasing with . Then is nonnegative and nonincreasing, and
In particular,
Equality in (48) at one positive time holds if and only if is a path of length and at one endpoint . Subdivision and reflection are immaterial. Thus the interval with all killing concentrated at one endpoint is the unique maximizer, as a metric space with killing measure, among the stated graphs and measures with fixed and .
For killing supported on finitely many vertices, the time integral of (48) is the already known sharp torsion inequality of Özcan and Täufer [OT, Theorem 8.2 in the cited preprint]. Their proof already places all strength at a minimum and uses energy rearrangement. Our additional assertion is the all-times concentration comparison (47), its extension to arbitrary finite measures, and rigidity from equality at one time. The key extra ingredient is a boundary inequality for the cumulative resolvent, rather than an energy comparison alone. Generalized couplings are also discussed in [BM, Remark 4.11]; that discussion does not supply (47). The fixed-graph ground-state maximization problem of Kurasov and Serio [KS] concerns a different functional and optimization direction.
7.1. Form and interval preliminaries
The compact embedding makes the measure term in (45) continuous on . The form norm after adding is equivalent to the norm, so the form is closed and has compact resolvent. It is a Dirichlet form: real normal contractions decrease both terms. Its semigroup and resolvents are therefore positivity preserving and sub-Markovian.
The spectrum is strictly positive. Indeed, choose with , using continuity and the minimum of . A simple path of length at most from to gives
In this section Lemma 2.1 has the following version without Dirichlet vertices: a continuous edgewise -Lipschitz function has a -Lipschitz decreasing rearrangement with . Use a simple path from a maximum to a minimum in the proof of (11); only levels between these two values are needed.
Fix and let . For nonnegative nonincreasing , the function is nonnegative and nonincreasing. For smooth , its derivative satisfies (14), now with ; the same positive-part argument applies. Monotone smooth approximation proves the assertion for data. For and one has
The differential equation follows by integration from the Neumann endpoint, and its value at , together with , gives the last condition. This is a Robin condition for with an inhomogeneous boundary value determined by the full integral of the datum.
We shall use the following elementary comparison. If satisfies
then integration against gives
Thus . In particular, (50) preserves cumulative order of decreasing data: if on , their solutions satisfy , including when .
7.2. Resolvent comparison for finitely many killing points
First suppose , with finitely many and . Insert degree-two vertices at interior atoms. The operator acts as on each open edge and satisfies
Here each derivative points from the vertex into its incident edge; this convention makes the derivative at the right endpoint of an interval equal to . Vertices without an atom have .
Take a strictly positive nonconstant polynomial load on each closed edge, and put . As in Lemma 3.2, is analytic and nonconstant on each edge. It is strictly positive everywhere. An interior zero contradicts . At a zero vertex all outgoing derivatives are nonnegative and sum to zero by (53); each is zero, and into each edge again contradicts nonnegativity. Thus .
For a level outside the finite set of critical and vertex values, use as in (19). There is at least one level preimage, by a path from a maximum to a minimum. Integrating over the superlevel set, with the vertex condition (53), gives
where and . In particular, the vertex term has a positive sign. Discard it and use and to obtain
The rearrangement is Lipschitz by Section 7.1, and only finitely many ranks are exceptional; thus this is also a weak inequality. The new boundary information follows by testing the full resolvent equation against the constant function:
Since every , (56) implies
Equations (55) and (57), compared to (50) with datum , give by (51)-(52). The polynomial density argument and the rearrangement contraction (10) pass this inequality to every nonnegative load. The last assertion following (52) then gives
for nonnegative and nonnegative decreasing .
7.3. General measures and evolution
Choose finite nonnegative atomic measures converging weakly to , all with mass ; finite partitions into sets of diameter tending to zero give such measures. On the fixed graph, the set of products with is relatively compact in . Weak convergence and the uniform bound therefore imply
For clarity, take a finite uniform -net of the compact closure of that product set. The integrals tend to zero on the net, and the error on the whole set is at most . Then let tend to zero.
The bilinear forms are uniformly coercive on , with coercivity constant at least for the usual norm. Subtracting their weak resolvent equations, testing with the difference, and applying (59) gives
for every . For example, if is the limiting resolvent solution, the norm of the difference is bounded by . Equation (9) now passes (58) to .
Iterate this resolvent comparison with . All interval iterates stay decreasing and preserve cumulative order by (50)-(52). Spectral dominated convergence, exactly as in (28), gives (47). Taking gives (48). This proves the inequality part of Theorem 7.1 without any regular-level assumption for the heat flow or for a singular measure.
7.4. Equality for an arbitrary killing measure
We give the additional details needed because strict edgewise concavity is generally false in the presence of a distributed potential.
First, every nonnegative satisfies the classical rearrangement inequality
One direct extension of the proof in Section 5 is to approximate in by nonnegative continuous piecewise linear functions on the edges. At every nonexceptional value their energy coarea formulas and prove (61); their finitely many constant pieces contribute zero energy on both sides. Rearrangement is continuous in by (10). The rearranged approximants are bounded in , so weak compactness and lower semicontinuity pass the estimate to . Continuity and connectedness also give whenever the rearrangement has its continuous representative. Hence
Suppose now that equality in (48) holds at . Set and . The half-time identities (35), with the corresponding forms, and Lemma 5.2 give . Differentiating the nonnegative heat-content deficit at its zero gives
The interval profile is strictly positive on and strictly decreasing on . To see strictness, the maximum-principle argument in Lemma 5.1 applies to the interval with its finite Robin endpoint: a zero there would have zero outward derivative and contradict the boundary point lemma. The semigroup is positivity improving, is sub-Markovian, and has strictly positive spectrum by (49). Thus on the open interval as in (31)-(32). Consequently and , so for . In particular, its distribution has no atoms, including at its endpoint values.
Equations (62)-(63) force equality separately in the gradient and killing terms:
Thus is supported on the minimum set of . We next show that this set is a single endpoint of a path.
Spectral smoothing gives . On each open edge the weak equation is
as distributions, with the measure restricted to that edge. The right side is a finite signed measure. Thus has a representative of bounded variation and is bounded on the edge; is edgewise Lipschitz. This assertion does not assert a sign for .
Put . Since , its distribution equals for , and is continuously differentiable there with . The one-dimensional area formula now gives, at almost every such value, finite level multiplicity and
Here we may exclude vertex values, images of nondifferentiability points and images of points where . These images have measure zero: Lipschitz maps preserve null sets, and the area formula makes the image of the zero-derivative set null. Moreover that set itself has length zero in this case. Its pushforward is supported on a null set of levels, whereas the distribution of , being that of , is absolutely continuous on and gives no mass to the endpoint levels. Thus the reciprocal-derivative formula loses no critical-set mass. Finite multiplicity almost everywhere follows from . These facts justify (66) without a finite-critical-points assertion.
The gradient equality in (64) and yield
Hence for almost every . No edge subinterval is constant, because that would create an atom in the distribution of . At a vertex of degree at least three choose disjoint small neighborhoods of its incidences. On each neighborhood takes a value different from the vertex value. By continuity, each incidence therefore covers an open interval of values either above or below the vertex value. Two of the three incidences cover overlapping intervals on the same side. This produces at least two distinct preimages on an open interval of levels, contrary to (67). All vertex degrees are at most two.
A finite connected graph with these degrees is a path or a cycle. On a cycle, two arcs from a minimum to a maximum give at least two preimages of every intermediate value, so a cycle is impossible. On a path, a continuous function with no constant subinterval and almost everywhere single level multiplicity must be strictly monotone: a failure of monotonicity produces a strict rise followed by a fall, or a strict fall followed by a rise, giving two preimages on an open interval of values. Its minimum is therefore attained at one endpoint only. The support conclusion from (64) gives there. Conversely, this measure on the path is the comparison problem, proving equality. Theorem 7.1 follows.
Corollary 7.2 (integrated comparison). Under (44),
with equality exactly in the case of Theorem 7.1. More generally, the positive-time-weight and convex-temperature conclusions of Section 6 hold with the Robin comparator.
Proof. The positive spectral gap makes the unweighted time integral finite. The interval torsion function is , whose integral is the right side of (68). Integrate (48) and use its strictness; the other conclusions follow as in Section 6.
For atomic vertex measures, (68) and its equality case are precisely prior work [OT, Theorem 8.2], not additional new results. Theorem 7.1 gives their all-times strengthening and allows general measures. The case is excluded: every graph then retains the constant temperature and has heat content , so rigidity fails. Quantitative stability and arbitrary vertex couplings remain outside the result. The two separate extremal comparisons do not settle the product problem discussed in [OT, Section 8.3]. Section 8 supplies the required joint argument for arbitrary finite nonnegative killing measures.
8. Kohler-Jobin inequalities for general killing measures
Let be a finite compact connected metric graph of length , without Dirichlet conditions, and let be any finite nonnegative Borel measure of mass . The form domain is the continuous space , as in Section 7. Set
For , let solve . The interval with Neumann condition at and killing has eigenvalue and torsion .
Theorem 8.1 (fixed length and total killing). Under these hypotheses,
Equality holds if and only if is a path and is concentrated at one endpoint, up to subdivision and reflection.
Theorem 8.2 (fixed torsion and total killing). Let satisfy . Then
Equivalently, with and ,
The equality case is the same as in Theorem 8.1. Both theorems include point interactions, integrable nonnegative potentials and singular continuous killing measures.
These results give sharp fixed-constraint formulations of the Robin/delta joint-optimization direction discussed in [OT, Section 8.3 of the cited preprint], extending the one-point statements of the preceding revision. The Dirichlet graph Kohler-Jobin theorem is due to Mugnolo and Plümer [MP, Theorem 5.8]; its nonlinear Dirichlet extension is considered in [O26]. Modified-torsion and level-set methods are established tools in this subject. Our proof uses a ground-state distribution functional in which Cauchy-Schwarz absorbs all killing terms. It then compares a scalar differential inequality with the cosine profile. Eigenfunction distribution inequalities of this kind belong to the classical Chiti comparison method; [DS, Lemma 3.2] treats Robin domains on the range of interior superlevel sets. Here connectedness and nonnegative killing give (79) over the full range of regular levels. The Euclidean domain Robin questions in [BCS, Section 5] impose different constraints and are not resolved here.
8.1. A ground-state distribution estimate
Define the continuous increasing function
Proposition 8.3. For every graph and measure above, with ,
Proof for finitely supported measures. Subdivide the graph at the atoms and write , where . Choose a positive first eigenfunction , and put and . Positivity follows by taking the absolute value in the form minimization and applying edgewise uniqueness: at a zero minimum the nonnegative outgoing derivatives sum to zero, and propagation along the connected graph would give the zero function. On each open edge, . Thus no edge piece is constant, there are finitely many critical and vertex values, and .
For , define
The positive minimum gives , so extends to a Lipschitz function on , with . Set . Cauchy-Schwarz gives . Testing the weak eigenvalue equation with therefore yields
Every test function here belongs to . The chain rules hold almost everywhere; the finitely many critical points cause no ambiguity. Cavalieri's formula and the torsion variational principle now give
In particular, killing away from a ground-state minimum is included in (76); it has not been dropped from the energy.
Let be the decreasing rearrangement on , and put . The function is positive and Lipschitz, with and . To justify Lipschitz continuity, let . A path crossing two levels spends length at least between them. Hence , which gives the asserted inverse bound. All level and inverse substitutions can also be made on the finitely many intervals of regular values and then passed to their endpoints.
For a regular , flux integration gives
Connectedness implies that a regular level has at least one preimage, so . Since the killing coefficients are nonnegative, . Using and inverting the distribution function gives
The interval in (77) has and . Substitution on the remaining levels, followed by integration by parts, yields
The term at cancels exactly, and at zero because . Including the interval below the positive minimum is essential.
Put , with . Since is decreasing, . Equation (79) gives
Consequently for . Thus lies in everywhere and satisfies on this initial interval. The function is nonnegative and increasing for . Combining (77)-(81) gives, with ,
The last identity follows by differentiation of (73). This proves (74) for finite atomic measures.
Passage to arbitrary measures. Choose finite atomic measures of mass on the same graph. The compact embedding and bounded total masses give
Indeed, products of two unit-ball functions form a relatively compact subset of , so weak convergence is uniform on that set by a finite-net argument. The coercivity estimate in Section 7 is uniform in . If and , subtracting the form equations and testing with therefore gives in and .
For completeness, the first eigenvalues converge as well. Testing with a normalized ground state of gives . Conversely, normalized ground states for have uniformly bounded norms; a subsequence converges weakly in and strongly in and . Its limit has norm one, and weak lower semicontinuity together with (83) gives . Thus . The continuity of the right side of (74) completes the proof.
8.2. Recovering the prescribed constraints and equality
We also need the familiar first-eigenvalue comparison
Here is a direct form proof that fixes the direction. For a nonnegative , the graph energy rearrangement proved in Section 7 and the minimum bound give
Taking absolute values first handles arbitrary , and minimization proves (84).
For , differentiation shows
because . For Theorem 8.1, put . Equation (84) gives . Therefore (74) and strict monotonicity give
If equality holds, and strict monotonicity force . Equality in (87) then gives . Corollary 7.2 characterizes the equality case as the endpoint-killed interval. Conversely, that interval attains equality.
For Theorem 8.2, let and . If , (74) is strictly stronger than (72). If , (84) and monotonicity of imply
Hence (74) gives , proving (72). The interval of length with strength has first eigenvalue and torsion . Interval torsion is strictly increasing with length, whereas its first eigenvalue is strictly decreasing; the latter also follows from . Thus (72) is equivalent to (71). Equality forces and again , giving the same rigidity by Corollary 7.2. This proves both theorems.
The two general-measure questions left open in revision 1.2 are therefore settled by (70)-(72). Quantitative stability of these inequalities and of the heat-content comparison remains open. Section 9 extends the stationary joint comparisons to nonlinear killing. Signed killing measures, discontinuous vertex couplings, weighted edge forms and the Euclidean Robin domain problem are outside the present assertions.
9. Nonlinear killing and the -Laplacian
The distribution argument extends to the full range . Fix a finite compact connected metric graph of length , without Dirichlet conditions, and a finite nonnegative Borel measure with . On the continuous space set
Let minimize , and distinguish its mass from the customary homogeneous torsional rigidity:
This convention for agrees with [O26]. In particular, the scale-invariant product is . The energy includes nonlinear point interactions as well as distributed killing. The results below concern stationary variational problems; no nonlinear heat-concentration theorem is asserted.
We specify the one-dimensional functions to avoid a normalization ambiguity. Define
Thus . For , let be the unique solution of
Theorem 9.1 (nonlinear joint comparisons). For every and every graph and measure above, put and . Then
Furthermore, if is chosen by
then . Equality in either comparison holds exactly when is a path and all killing is at one endpoint, up to subdivision and reflection.
The nonlinear Dirichlet graph Kohler-Jobin inequality and its normalization are due to [O26, Theorem 5.2], following the nonlinear modified-torsion strategy of Kohler-Jobin and Brasco. Its level-set flux inequality and modified-torsion test are established ingredients [O26, Lemma 5.3]. Here the Hölder composition estimate below includes all finite killing terms, and the scalar comparison retains the finite Robin parameter. Thus the new assertion concerns nonnegative measure killing; the Dirichlet theorem is credited, not claimed anew.
9.1. Variational facts and interval rigidity
We first justify the variational objects and the separate interval comparisons. Path integration and Hölder's inequality give, uniformly over measures of mass ,
Indeed, ; raise to the power and integrate over with length measure and over with . Consequently controls the norm. The compact embedding into , the direct method and strict convexity give a unique torsion minimizer , a positive minimum eigenvalue and a nonnegative normalized ground state. Strict convexity follows because equality in the gradient term makes two functions differ by a constant, while the positive mass of excludes a nonzero constant difference.
The torsion Euler equation and its variational value are
Testing with gives . The minimizer is not zero, since a small positive constant makes negative. Optimizing scalar multiples of a test function also gives .
For the endpoint-killed interval, direct integration gives
To compute its eigenvalue, put . Differentiation of (91) gives and
A positive interval ground state decreases from its Neumann endpoint; its flux equation and initial value therefore identify it, after normalization, with , where . Its Robin condition is , so
Lemma 9.2 (separate comparisons). One has and . Equality in the torsion comparison holds exactly for the endpoint-killed interval.
Proof. The graph rearrangement inequality extends to -energy:
For a continuous piecewise affine function, coarea and Hölder give , where and . Integrating gives the gradient inequality. Piecewise affine approximation, the contraction of rearrangement and weak lower semicontinuity extend it to , including constant pieces. The killing term is at least , and the continuous representative of has endpoint value . This proves (100). The Rayleigh and torsion variational principles give the two comparisons.
If , uniqueness of the interval torsion minimizer gives . Both nonnegative energy defects in (100) vanish, and is supported where . We explain why the graph is a path even for singular measures. On each edge the distributional equation is
The flux is of bounded variation, hence bounded; therefore is bounded and is Lipschitz. Its rearrangement is strictly decreasing, with nonzero derivative on . Its distribution is absolutely continuous between endpoint values and has no atoms. The one-dimensional area formula makes the image of the critical and nondifferentiability set of null. Absolute continuity of the distribution then shows that this set has zero length. Thus the coarea energy identities and inverse differentiation apply without losing mass, as in Section 7.4. Equality in (100) forces at almost every level.
There are no constant edge pieces, since they would create atoms in the distribution. At a vertex with at least three incidences, two disjoint incident germs have overlapping value intervals on the same side of the vertex value; those levels have at least two preimages. A cycle likewise has two arcs joining a minimum to a maximum. Hence the connected graph has no branch and no cycle, and is a path. A nonmonotone continuous function on a path would have repeated levels on an interval, so is strictly monotone. Its minimum is one endpoint, where all of is concentrated. Formula (97) proves the converse.
9.2. The distribution bound with nonlinear killing
Define, continuously at the endpoints,
Proposition 9.3. With ,
Proof for atomic measures. Subdivide at every atom. A nonnegative first eigenfunction is strictly positive. Indeed, at a zero minimum every outgoing flux is nonnegative and their sum is zero. On an adjacent edge, the conserved quantity is then zero, forcing that edge solution to vanish; connectedness propagates this contradiction. The same argument excludes an interior zero. On each edge the flux derivative is , so the derivative is strictly decreasing and there is at most one critical point. In particular there are no constant edge pieces. Write , , and
Since , the function is Lipschitz on . Put . Hölder gives . Testing the weak eigenvalue equation with , and using the nonnegativity of , gives
These compositions are in and all chain rules are justified by their Lipschitz bounds and the finite exceptional set. Equations (96) and (105), together with Cavalieri's formula, yield
Indeed the integrand before maximizing is , and (104) attains its pointwise maximum.
Let and . The path argument of Section 8 makes positive and Lipschitz. On regular levels, total flux, including killing, equals . Hölder's coarea inequality therefore implies
Including the levels in (106), changing to the rank variable and integrating by parts gives
The term at cancels the first term. The term at zero vanishes because .
Put , so , and . From (107),
For , integration of gives . Substituting in this integral and using (91) shows that, for ,
The right side belongs to : the model ratio is at least the rank variable, or equivalently , which follows directly from (91).
The function is nonnegative and increasing on . Differentiating (102), using , gives
The endpoint values in (102) follow from (91), in particular as . Equations (106)-(111), with , imply
Only the interval before the pole of is compared. The remaining integrand, when present, is nonnegative. This proves the atomic case.
Passage to arbitrary measures. Approximate weakly on the fixed graph by finite atomic measures of the same mass . Compactness of the unit ball in implies uniform convergence of to on bounded sets. The bound (95) is uniform in . Testing Rayleigh quotients with a limiting ground state gives the eigenvalue limsup inequality; compactness of normalized minimizing ground states, convergence in and weak lower semicontinuity of the gradient energy give the liminf inequality. Thus .
For torsion, the minimizing values of are uniformly coercive by (95) and Young's inequality. Every subsequence of minimizers has a further subsequence converging weakly in and strongly in . Lower semicontinuity and comparison with any fixed test function show that the limit minimizes . Uniqueness identifies it with , so . Continuity of the right side of (103), including at , completes the proof.
9.3. Proof of the joint comparisons
Lemma 9.2 and (99) give . Since is strictly increasing,
By (92), , so (113) is (93). Equality forces and ; Lemma 9.2 gives the stated rigidity.
For the second comparison put and . If , (103) gives . Otherwise (99) and the monotonicity of imply , hence . Therefore in all cases
The interval of length with killing mass has eigenvalue and torsion mass . At fixed , (97) strictly increases with length, while the eigenvalue strictly decreases, as is also seen from . Thus (114) is equivalent to the comparison with the interval in (94). Equality forces and again , proving rigidity. The endpoint interval attains equality in both statements.
At , all definitions and constants reduce exactly to Section 8. As , the sharp constant in (93) tends to , the nonlinear Dirichlet graph constant in [O26, equation (5.4)]. Quantitative stability, nonlinear heat concentration and signed-measure analogues remain outside these results.
References
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This is an AI-assisted research preprint prepared under Henry Zweiman's name at his request. The proof and the primary-source comparison were audited internally by the originating assistant; no independent expert review or journal acceptance is claimed. The accompanying assessment and source records state the search scope, the source passages inspected and the remaining priority limitations. The mathematical statements above are supported by the displayed arguments, rather than by finite computations or artifact checks.