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Nontrivial solutions for a class of semilinear elliptic equations with a nonlinear Goldstein-Wentzell boundary condition

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Pith's one-line read This paper proves that a doubly elliptic problem with a nonlinear Goldstein–Wentzell boundary condition has nontrivial lowest-energy solutions exactly at the potential-well depth, and infinitely many higher-energy solutions when the sources

desk verdict A solid, careful extension of Vitillaro's potential-well program to bulk–surface problems with two independent nonlinearities; the central characterization d=c=inf_N I looks right, with the main caveat being reliance on cited prior theorems. read the letter →

arxiv 2607.28709 v1 pith:W7VSDTNO submitted 2026-07-30 math.AP math-phmath.MP

Enzo Vitillaro This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35D3035J0535J2035J2535J6135J67
keywords Goldstein–Wentzellboundaryconditionsemilinearellipticequationpotential-welldepthNeharimanifoldMountainPassTheoremLaplace–Beltramioperatorleastenergysolutionsmultiplicityof
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a semilinear elliptic equation in a bounded domain, with a nonlinear boundary condition that combines the Laplace–Beltrami operator with the normal derivative on part of the boundary (a nonlinear Goldstein–Wentzell condition). Its main claim is that, under subcritical growth, monotonicity, and a strict-monotonicity condition on the ratio of the sources to u, the problem always has a nontrivial weak solution whose energy is exactly the depth d of the potential well. The paper proves that this depth is simultaneously the Mountain Pass level c and the least energy on the Nehari manifold, so any solution at level d is a lowest-energy nontrivial solution. It also shows that when the two nonlinearities are odd, there are infinitely many nontrivial solutions with energies tending to infinity, and that in the purely power-like case the threshold d is given by an explicit formula involving the best Sobolev embedding constant. A careful reader should care because this identifies the threshold energy that separates global existence from blow-up in the associated hyperbolic boundary-value problem, and gives computable values for it.

What carries the argument

The key object is the energy functional I(u) = 1/2∫_Ω |∇u|² + 1/2∫_Γ1 |∇_Γ u|² − ∫_Ω F(u) − ∫_Γ1 G(u), defined on the space H^1 of functions in H^1(Ω) whose trace lies in H^1(Γ) and vanishes on Γ0. Its potential-well depth is d = inf_{u≠0} sup_{t>0} I(tu). The paper shows that d = c = inf_N I, where c is the Mountain Pass level and N = {u≠0 : K(u)=0} is the Nehari manifold. The engine of the proof is the one-variable ray analysis in Lemma 3.5: the derivative of θ_u(t)=I(tu) has at most one zero because the functions σ(u)=h(u)/u and ξ(u)=k(u)/u are monotone and, by assumption (A4), have no flat positive plateaus; the mountain pass path can therefore be taken along the ray itself. This same st

What would settle it

Compute the least-energy critical level in the homogeneous case f(u)=γ|u|^{p−2}u, g≡0 on a domain with explicitly known Sobolev embedding constant B_Ω (e.g., an interval or a ball), and compare the energy of the minimizer with the predicted value (1/2−1/p)γ^{−2/(p−2)}B_Ω^{−2p/(p−2)}. Any disagreement falsifies Theorem 1.4 and, through it, Theorems 1.1–1.2.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: under assumptions (A1)–(A4) the problem (1.1) has at least one nontrivial weak solution u ∈ H^1 with energy I(u) = d > 0, and this d coincides with the Mountain Pass level c of the energy functional. Theorem 1.2 adds that the same d is the infimum of I over the Nehari manifold N, so solutions at level d are exactly the lowest-energy nontrivial solutions; conversely, any u satisfying u ∈ N and I(u) ≤ d solves (1.1). Theorem 1.3 states that for odd f and g there is a sequence of solutions (u_n) with I(u_n) = I(−u_n) → ∞. Theorems 1.4–1.6 compute d explicitly in the homogeneous power cases, relating it to the best constant of the Sobolev embedding H^1 → L^p(Ω),

Load-bearing premise

The load-bearing premise is that the ratio functions σ(u)=h(u)/u and ξ(u)=k(u)/u never stay constant at a positive level; if one of them has a positive flat plateau, the proof that each direction has a unique ray maximum and that the Nehari manifold is a regular constraint would no longer go through, and the equality d=c=inf_N I would not follow.

Editorial extensions

If this is right

  • Nontrivial stationary solutions of the associated hyperbolic boundary-value problem exist exactly at the threshold energy d, which is the minimal possible energy of a nontrivial stationary solution.
  • The threshold d is simultaneously the mountain-pass level and the Nehari minimum, so any variational method that finds a critical point at one of these levels automatically produces a lowest-energy nontrivial solution.
  • In the homogeneous power case the threshold has a closed form, e.g., d=(1/2−1/p)γ^{−2/(p−2)}B_Ω^{−2p/(p−2)}, so the energy level can be computed from the best Sobolev embedding constant.
  • In the boundary-only case f≡0 with g(u)=δ|u|^{q−2}u, the formula reduces to the threshold computed in the author's earlier special case, giving a continuous extension of the previously known result.
  • When f and g are odd, the problem has infinitely many solutions with unbounded energy, both in the interior-dominated and boundary-dominated regimes; in the boundary-dominated regime the same conclusion holds after reduction to an auxiliary Dirichlet problem via the Dirichlet-to-Neumann operator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since the only place the 'no critical points' condition (A4) enters is the strict monotonicity of the ray function, the equality d=c=inf_N I is likely to fail for nonlinearities where h(u)/u or k(u)/u has a positive flat plateau; the theorems should be read as sharp for the class they cover.
  • The explicit formulas in Theorems 1.4–1.6 give a practical way to estimate the blow-up threshold in the dynamical problem: only the best constant of a Sobolev or trace embedding needs to be computed, which can be done numerically for a given domain.
  • The same ray-monotonicity + Nehari-constraint argument could be transferred to other boundary conditions (e.g., p-Laplacian, or acoustic boundary conditions) where a single positive threshold separates global existence from blow-up.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper proves existence, characterization, and multiplicity results for a doubly elliptic problem with a nonlinear Goldstein–Wentzell boundary condition: an interior semilinear equation coupled with a Laplace–Beltrami boundary condition and independent boundary source g(u). Under hypotheses (A1)–(A4), Theorem 1.1 gives a nontrivial weak solution at the potential-well depth d, and shows d coincides with the Mountain–Pass level c. Theorem 1.2 identifies d with the infimum of the associated Nehari functional and characterizes level-d solutions as lowest-energy nontrivial solutions. Theorem 1.3 establishes infinitely many pairs of solutions when f and g are odd. Theorems 1.4–1.6 compute d explicitly in the positively homogeneous, odd cases and relate level-d solutions to Sobolev/trace embedding maximizers. The proof strategy is a variational one: the energy functional I on the space H^1 is shown to satisfy the Palais–Smale condition, a ray-by-ray analysis gives the potential-well structure, and a variant of the Mountain Pass Theorem yields the critical point.

Significance. If correct, the paper gives a coherent variational picture for a problem with two independent source terms, one in the interior and one on the boundary, and it allows linear behavior near the origin together with asymmetric one-sided limits. The central equalities d=c=inf_N I are derived from the assumptions rather than assumed. The paper is careful in verifying the Palais–Smale condition via the uniform estimate (3.26), in checking the Nehari constraint through Lemma 3.6, and in treating the asymmetry cases in Lemma 3.5. The homogeneous results in Theorems 1.4–1.6 give explicit, parameter-free formulas for d in terms of embedding constants, and the converse statements characterize the maximizers. I found no circularity and no fitted parameters. The main caveat is that the key Mountain–Pass variant, Theorem 2.2, is cited from the author's earlier work [59] rather than proved here; this reduces self-containedness but is not, by itself, a correctness defect.

minor comments (6)
  1. [§3.2, Eq. (3.28)] The definition of the set S contains ambiguous notation. In the second and third cases, "lim_{u→∞} σ(u)>0 = lim_{u→−∞} σ(u)" appears to mean that one limit is positive and the other is zero, but as written it duplicates the first case. Please rewrite, e.g., "lim_{u→∞} σ(u)>0 and lim_{u→−∞} σ(u)=0" and symmetrically for the third case. This classification is used in Lemma 3.5, so clarity matters.
  2. [§3.1, Lemma 3.2] The sentence near (3.9) stating that dI: H^1 → (H^1)' is compact is incorrect: dI is the Riesz isomorphism minus the compact operator dJ, and the Riesz map is not compact. What is proved and used later is the compactness of dJ, not of dI. Please correct the wording.
  3. [§3.1, Lemma 3.4] The notation "τ0 := min{p2, q0}" should presumably be "min{p0, q0}". As written, p2 is undefined. The subsequent inequalities use p0 and q0, so this is a typographical slip, but it should be fixed.
  4. [Proof of Theorem 1.2] In the reverse-inequality part, the reference "By (1.31)–(1.32) and (3.41)" is incorrect: (1.31)–(1.32) are statements from Theorem 1.4 and are not relevant here. The intended reference is likely (3.44) together with (3.41), since K(u)=0 gives θ'_u(1)=0. Please correct the cross-reference.
  5. [Theorem 1.5, Eq. (1.37)] The second norm in (1.37) should be \|u\|_{L^q(Γ_1)}, not \|u\|_{L^q(Ω)}. This is clear from the context and from Lemma 5.2, but the statement as printed is a typographical error.
  6. [§2.4, Theorem 2.2] Theorem 2.2 is the central existence tool for Theorem 1.1, but it is cited from [59] without proof. I am not objecting to the citation, but for the reader's convenience and for independent verification, please state explicitly that the hypotheses of [59, Theorem 5] match exactly the present setting, or include a short proof in an appendix. This is a verifiability request rather than a mathematical objection.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the equalities d=c=inf_N I are proved from assumptions and standard critical point theory, not by construction.

full rationale

The paper's central claim is that the potential-well depth d (1.19), the Mountain-Pass level c (1.20), and the Nehari minimum inf_N I (1.23) coincide. These are distinct definitions, and the proof does not identify them trivially. Lemma 3.5 proves under (A1-4) that each ray has a unique maximizer unless it lies in S, and Proposition 3.1 obtains a critical point at level c via the Mountain-Pass variant Theorem 2.2. The inequalities d>=c and d<=c are then shown explicitly in the proof of Theorem 1.1. Theorem 1.2 uses the Nehari identity K(v)=<dI(v),v> and Lemma 3.6; it does not redefine d. There are no fitted parameters and no quantity called a prediction that is secretly an input. The paper does rely on the author's previous work: Theorem 2.2 is cited with proof in [59, Proof of Theorem 5], the Rayleigh formula (3.11) is cited from [60, Theorem 1.2], and Lemma 2.1 is cited from [59]-[62]. However, these are general theorems with stated assumptions that do not contain the target problem, so by rule 4 they count as independent support rather than circularity. The explicit statement that the delta=1 case of Theorem 1.5 reduces to [59, Theorem 2] is an acknowledged consistency check, not a renaming. No self-definitional step, fitted-input step, or author-uniqueness argument can be exhibited with the required specificity. The score 2 reflects the presence of several self-citations, but the central derivation is internally coherent and independent of its own conclusions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard critical point theory, Sobolev/trace embedding theorems, and prior eigenvalue results by the author. There are no fitted parameters and no invented physical entities. The only truly ad hoc element is assumption (A4), which is explicitly imposed to make the ray and Nehari arguments work.

assumptions (7)
  • standard math First eigenvalue λ1 of the doubly elliptic problem (1.5) is positive and satisfies the Rayleigh formula (3.11).
    Cited from [60, Theorems 1.1–1.2] and [61]; used throughout to obtain coercivity (3.12) and the key comparison a0<λ1.
  • standard math Sobolev and trace embeddings H^1(Ω)↪L^p(Ω) and H^1(Γ1)↪L^q(Γ1) are continuous and compact for subcritical p,q.
    Used in Lemma 3.2 and in estimates (3.14)–(3.15); stated as classical background.
  • standard math Mountain Pass Theorem, Z2-symmetric Mountain Pass Theorem, and the Nehari-manifold lemma from [4, Proposition 6.7].
    Theorem 2.1–2.3 are cited from [46] and [4]; Lemma 3.6 refines [4, Proposition 6.7]; these tools drive Theorems 1.1–1.3.
  • domain assumption The weak-solution distribution identity for the evolutionary problem (1.2) from [57, Lemma 3.3].
    Used in Lemma 3.1 to identify stationary weak solutions of (1.2) with weak solutions of (1.1); relies on assumption (1.13).
  • domain assumption Standing geometric hypotheses: Ω connected, Γ of class C^1, Γ0∩Γ1 empty in the relevant sense, H^{N-1}(Γ0)>0.
    Stated at the start of §1.1; used for λ1>0 and for the norm equivalence in Lemma 2.1. Failure can break coercivity.
  • standard math Lax–Milgram solvability of the auxiliary nonhomogeneous Dirichlet problem (4.4) when λ<λ1^D.
    Used in Lemma 4.1 and Lemma 4.2 to construct the Dirichlet-to-Neumann map D in the case B) of the multiplicity proof.
  • ad hoc to paper Assumption (A4): σ and ξ have no critical points at positive level.
    Used in Lemma 3.5 to force a unique maximum along rays and in Theorem 1.2 to prove ⟨dK(u),u⟩≠0 on the Nehari manifold. This is the most restrictive new hypothesis.

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Cite this review

Pith. "Pith review of Nontrivial solutions for a class of semilinear elliptic equations with a nonlinear Goldstein-Wentzell boundary condition." pith.science (2026). https://pith.science/paper/W7VSDTNO

@misc{pith2026260728709,
  author       = {Pith},
  title        = {Pith review of: Nontrivial solutions for a class of semilinear elliptic equations with a nonlinear Goldstein-Wentzell boundary condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7VSDTNO}},
  note         = {Machine review of arXiv:2607.28709}
}
abstract

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} -\Delta u=f(u) \qquad &\text{in $\Omega$,}\\ \phantom{-}u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =g(u)\qquad &\text{on $\Gamma_1$,} \end{cases} $$ where $\Omega$ is a bounded open domain of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\Gamma=\partial\Omega$, with $\Gamma=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $\Gamma_1$ being nonempty and relatively open on $\Gamma$, $\mathcal{H}^{N-1}(\Gamma_0)>0$. The terms $f$ and $g$ are subcritical with respect to Sobolev embeddings, respectively in $\Omega$ and on $\partial\Omega$. We prove that, under suitable assumptions, the problem admits nontrivial solutions at the depth of the potential well energy level, which is the minimum energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.

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Reference graph

Works this paper leans on

68 extracted references

  1. [59]

    ,Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition, Commun. Anal. Mech.15(2023), no. 4, 811–830

    2023

  2. [1]

    R. A. Adams,Sobolev spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, Vol. 65

    1975

  3. [2]

    Adimurthi and S. L. Yadava,Positive solution for Neumann problem with critical nonlinear- ity on boundary, Comm. Partial Differential Equations16(1991), no. 11, 1733–1760

    1991

  4. [3]

    Ambrosetti,Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro, Atti Accad

    A. Ambrosetti,Esistenza di infinite soluzioni per problemi non lineari in assenza di parametro, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8)52(1972), 660–667

    1972

  5. [4]

    Ambrosetti and A

    A. Ambrosetti and A. Malchiodi,Nonlinear analysis and semilinear elliptic problems, Cam- bridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007

    2007

  6. [5]

    Ambrosetti and G

    A. Ambrosetti and G. Prodi,A primer of nonlinear analysis, Cambridge University Press, Cambridge, 1993

    1993

  7. [6]

    Ambrosetti and P

    A. Ambrosetti and P. H. Rabinowitz,Dual variational methods in critical point theory and applications, J. Functional Analysis14(1973), 349–381

    1973

  8. [7]

    Atkinson, D

    K. Atkinson, D. Chien, and O. Hansen,A spectral method for an elliptic equation with a nonlinear Neumann boundary condition, Numer. Algorithms81(2019), no. 1, 313–344. MR 3943635

    2019

Show all 68 references
  1. [8]

    Ben Ayed, H

    M. Ben Ayed, H. Fourti, and A. Selmi,Harmonic functions with nonlinear Neumann bound- ary condition and their Morse indices, Nonlinear Anal. Real World Appl.38(2017), 96–112

    2017

  2. [9]

    Bociu,Local and global wellposedness of weak solutions for the wave equation with nonlin- ear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal

    L. Bociu,Local and global wellposedness of weak solutions for the wave equation with nonlin- ear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal. 71(2009), no. 12, e560–e575

    2009

  3. [10]

    Bociu and I

    L. Bociu and I. Lasiecka,Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations249(2010), no. 3, 654–683

    2010

  4. [11]

    J. F. Bonder and J. D. Rossi,Existence results for thep-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl.263(2001), no. 1, 195–223

    2001

  5. [12]

    W. M. Boothby,An introduction to differentiable manifolds and Riemannian geometry, Aca- demic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, No. 63

    1975

  6. [13]

    M. M. Cavalcanti, V. N. Domingos Cavalcanti, and I. Lasiecka,Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping—source interaction, J. Differential Equations236(2007), no. 2, 407–459

    2007

  7. [14]

    M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez,Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Dif- ferential Equations203(2004), no. 1, 119–158

    2004

  8. [15]

    Chipot, I

    M. Chipot, I. Shafrir, and M. Fila,On the solutions to some elliptic equations with nonlinear Neumann boundary conditions, Adv. Differential Equations1(1996), no. 1, 91–110

    1996

  9. [16]

    Chleb ´ ık, M

    M. Chleb ´ ık, M. Fila, and W. Reichel,Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition, NoDEA Nonlinear Differential Equations Appl.10(2003), no. 3, 329–346

    2003

  10. [17]

    Chueshov, M

    I. Chueshov, M. Eller, and I. Lasiecka,On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations 27(2002), no. 9-10, 1901–1951

    2002

  11. [18]

    G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht, and S. Romanelli,The role of Wentzell boundary conditions in linear and nonlinear analysis, Advances in nonlinear analysis: theory methods and applications, Math. Probl. Eng. Aerosp. Sci., vol. ...

    2009

  12. [19]

    Dambrine, D

    M. Dambrine, D. Kateb, and J. Lamboley,An extremal eigenvalue problem for the Wentzell- Laplace operator, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire33(2016), no. 2, 409–450

    2016

  13. [20]

    del Pino and C

    M. del Pino and C. Flores,Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations26(2001), no. 11- 12, 2189–2210

    2001

  14. [21]

    F. Du, Q. Wang, and C. Xia,Estimates for eigenvalues of the Wentzell-Laplace operator, J. Geom. Phys.129(2018), 25–33

    2018

  15. [22]

    C. M. Elliott and T. Ranner,Finite element analysis for a coupled bulk-surface partial dif- ferential equation, IMA J. Numer. Anal.33(2013), no. 2, 377–402. NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR... 39

    2013

  16. [23]

    L. C. Evans,Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010

    2010

  17. [24]

    Favini, C

    A. Favini, C. G. Gal, G. Ruiz Goldstein, J. A. Goldstein, and S. Romanelli,The non- autonomous wave equation with general Wentzell boundary conditions, Proc. Roy. Soc. Ed- inburgh Sect. A135(2005), no. 2, 317–329

    2005

  18. [25]

    Fiscella and E

    A. Fiscella and E. Vitillaro,Blow-up for the wave equation with nonlinear source and bound- ary damping terms, Proc. Roy. Soc. Edinburgh Sect. A145(2015), no. 4, 759–778

    2015

  19. [26]

    ,Blow-up for the wave equation with nonlinear source and boundary damping terms – corrigendum, Proc. Roy. Soc. Edinburgh Sect. A (2026), 1–4

    2026

  20. [27]

    Girault and P.-A

    V. Girault and P.-A. Raviart,Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986, Theory and algorithms

    1986

  21. [28]

    Ruiz Goldstein,Derivation and physical interpretation of general boundary conditions, Adv

    G. Ruiz Goldstein,Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations11(2006), no. 4, 457–480

    2006

  22. [29]

    Greco and G

    A. Greco and G. Viglialoro,Existence and uniqueness for a two-dimensional Ventcel problem modeling the equilibrium of a prestressed membrane, Appl. Math.68(2023), no. 2, 123–142

    2023

  23. [30]

    Grisvard,Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol

    P. Grisvard,Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, Reprint of the 1985 original, with a foreword by Susanne C. Brenner

    2011

  24. [31]

    Hebey,Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lec- ture Notes in Mathematics, vol

    E. Hebey,Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lec- ture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999

    1999

  25. [32]

    Inkmann,Existence and multiplicity theorems for semilinear elliptic equations with non- linear boundary conditions, Indiana Univ

    F. Inkmann,Existence and multiplicity theorems for semilinear elliptic equations with non- linear boundary conditions, Indiana Univ. Math. J.31(1982), no. 2, 213–221

    1982

  26. [33]

    Jost,Riemannian Geometry and Geometric Analysis, fifth ed., Universitext, Springer- Verlag, Berlin, 2008

    J. Jost,Riemannian Geometry and Geometric Analysis, fifth ed., Universitext, Springer- Verlag, Berlin, 2008

    2008

  27. [34]

    Kashiwabara, C

    T. Kashiwabara, C. M. Colciago, L. Ded` e, and A. Quarteroni,Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem, SIAM J. Numer. Anal.53(2015), no. 1, 105–126

    2015

  28. [35]

    Kato,Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition

    T. Kato,Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition

    1995

  29. [36]

    Knopf and C

    P. Knopf and C. Liu,On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: well-posedness, regularity theory and eigenvalue problems, Interfaces Free Bound.23(2021), no. 4, 507–533

    2021

  30. [37]

    Lasiecka and D

    I. Lasiecka and D. Tataru,Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equation6(1993), no. 3, 507–533

    1993

  31. [38]

    J. Li, L. Su, X. Wang, and Y. Wang,Bulk-surface coupling: derivation of two models, J. Differential Equations289(2021), 1–34

    2021

  32. [39]

    Lions,Lectures on elliptic partial differential equations, Tata Institute of Fundamental Research Lectures on Mathematics, vol

    J.-L. Lions,Lectures on elliptic partial differential equations, Tata Institute of Fundamental Research Lectures on Mathematics, vol. No. 10, Tata Institute of Fundamental Research, Bombay, 1967

    1967

  33. [40]

    Mugnolo and E

    D. Mugnolo and E. Vitillaro,The wave equation with acoustic boundary conditions on non- locally reacting surfaces, Mem. Amer. Math. Soc.303(2024), no. 1526

    2024

  34. [41]

    Nicaise, H

    S. Nicaise, H. Li, and A. Mazzucato,Regularity and a priori error analysis of a Ventcel problem in polyhedral domains, Math. Methods Appl. Sci.40(2017), no. 5, 1625–1636

    2017

  35. [42]

    L. E. Payne and D. H. Sattinger,Saddle points and instability of nonlinear hyperbolic equa- tions, Israel J. Math.22(1975), 273–303

    1975

  36. [43]

    Pierotti and S

    D. Pierotti and S. Terracini,On a Neumann problem with critical exponent and critical nonlinearity on the boundary, Comm. Partial Differential Equations20(1995), no. 7-8, 1155– 1187

    1995

  37. [44]

    Pucci and E

    P. Pucci and E. Vitillaro,Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems, Boll. Unione Mat. Ital.13(2020), no. 4, 487–494. MR 4172949

    2020

  38. [45]

    Quittner and W

    P. Quittner and W. Reichel,Very weak solutions to elliptic equations with nonlinear Neu- mann boundary conditions, Calc. Var. Partial Differential Equations32(2008), no. 4, 429– 452

    2008

  39. [46]

    P. H. Rabinowitz,Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Conference Board 40 ENZO VITILLARO of the Mathematical Sciences, Washington, DC; by the American Mathematical Society...

    1986

  40. [47]

    Romanelli,Goldstein-Wentzell boundary conditions: recent results with Jerry and Gis` ele Goldstein, Discrete Contin

    S. Romanelli,Goldstein-Wentzell boundary conditions: recent results with Jerry and Gis` ele Goldstein, Discrete Contin. Dyn. Syst.34(2014), no. 2, 749–760

    2014

  41. [48]

    D. H. Sattinger,Stability of nonlinear hyperbolic equations, Arch. Rational Mech. Anal.28 (1968), 226–244

    1968

  42. [49]

    Sternberg,Lectures on differential geometry, second ed., Chelsea Publishing Co., New York, 1983, With an appendix by Sternberg and Victor W

    S. Sternberg,Lectures on differential geometry, second ed., Chelsea Publishing Co., New York, 1983, With an appendix by Sternberg and Victor W. Guillemin

    1983

  43. [50]

    M. E. Taylor,Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer- Verlag, New York, 1996, Basic theory

    1996

  44. [51]

    Terracini,Symmetry properties of positive solutions to some elliptic equations with non- linear boundary conditions, Differential Integral Equations8(1995), no

    S. Terracini,Symmetry properties of positive solutions to some elliptic equations with non- linear boundary conditions, Differential Integral Equations8(1995), no. 8, 1911–1922

    1995

  45. [52]

    J. L. Vazquez and E. Vitillaro,Wave equation with second-order non-standard dynamical boundary conditions, Math. Models Methods Appl. Sci.18(2008), no. 12, 2019–2054

    2008

  46. [53]

    J. L. V´ azquez and E. Vitillaro,Heat equation with dynamical boundary conditions of reactive- diffusive type, J. Differential Equations250(2011), no. 4, 2143–2161

    2011

  47. [54]

    Vitillaro,Global existence for the wave equation with nonlinear boundary damping and source terms, J

    E. Vitillaro,Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations186(2002), no. 1, 259–298

    2002

  48. [55]

    ,Strong solutions for the wave equation with a kinetic boundary condition, Recent trends in nonlinear partial differential equations. I. Evolution problems, Contemp. Math., vol. 594, Amer. Math. Soc., Providence, RI, 2013, pp. 295–307

    2013

  49. [56]

    ,On the Wave Equation with Hyperbolic Dynamical Boundary Conditions, Interior and Boundary Damping and Source, Arch. Ration. Mech. Anal.223(2017), no. 3, 1183–1237

    2017

  50. [57]

    Differential Equations265(2018), no

    ,On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, J. Differential Equations265(2018), no. 10, 4873–4941

    2018

  51. [58]

    ,Blow-up for the wave equation with hyperbolic dynamical boundary conditions, in- terior and boundary nonlinear damping and sources, Discrete Contin. Dyn. Syst. Ser. S14 (2021), no. 12, 4575–4608

    2021

  52. [60]

    ,On the eigenvalue problem for a bulk/surface elliptic system, Commun. Math. Anal. Appl.4(2025), no. 3, 307–335. MR 4958220

    2025

  53. [61]

    1, 154–156

    ,Corrigendum to: On the eigenvalue problem for a bulk/surface elliptic system, Com- munications in Mathematical Analysis and Applications5(2026), no. 1, 154–156

    2026

  54. [62]

    ,On the necessity of the connectedness condition ofΩin : ”Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition”, Commun. Anal. Mech.18(2026), no. 2, 400–405

    2026

  55. [63]

    Differential Equations252(2012), no

    Tsung-fang Wu,Existence and multiplicity of positive solutions for a class of nonlinear boundary value problems, J. Differential Equations252(2012), no. 5, 3403–3435

    2012

  56. [64]

    Xia and Q

    C. Xia and Q. Wang,Eigenvalues of the Wentzell-Laplace operator and of the fourth order Steklov problems, J. Differential Equations264(2018), no. 10, 6486–6506

    2018

  57. [65]

    Xiao and L

    T.-J. Xiao and L. Jin,Complete second order differential equations in Banach spaces with dynamic boundary conditions, J. Differential Equations200(2004), no. 1, 105–136

    2004

  58. [66]

    Xiao and J

    T.-J. Xiao and J. Liang,Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc.356(2004), no. 12, 4787–4809

    2004

  59. [67]

    Zhang,Stabilization of the wave equation with variable coefficients and a dynamical bound- ary control, Electronic Journal of Differential Equations2016(2016), no

    Z. Zhang,Stabilization of the wave equation with variable coefficients and a dynamical bound- ary control, Electronic Journal of Differential Equations2016(2016), no. 27, 1–10

    2016

  60. [68]

    Zuazua,Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J

    E. Zuazua,Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim.28(1990), 466–477. (E. Vitillaro)Dipartimento di Matematica e Informatica, Universit `a di Perugia, Via V anvitelli,1 06123 Perugia ITALY Email address:enzo.vitillaro@unipg.it

    1990

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