arXiv subjects

Vladimir Bobkov

Publications and source records attributed to Vladimir Bobkov.

At least 19 recordsLinked to original sources

arXiv2026

Remarks on the antimaximum principle

We present several observations on the antimaximum principle (AMP) for the model problem $-\Delta_p u = \lambda |u|^{p-2} u + f$ in a bounded smooth domain $\Omega$, subject to the zero Dirichlet boundary conditions, and where the source function $f$ is nontrivial, nonnegative, and sufficiently regular. Denote by $\lambda_f$ the endpoint of validity of the AMP, so that every solution of the problem is negative in $\Omega$ for any $\lambda \in (\lambda_1,\lambda_f)$. Our discussion covers the following aspects: identification of a class of sources over which the AMP is uniform, lower semicontinuity of the map $f \mapsto \lambda_f$, bounds on $\lambda_f$, the nonexistence of negative solutions for sufficiently large $\lambda$ (extended AMP), the anticomparison principle, and the weakening of the source regularity from the Lebesgue to Morrey spaces. Some of the results are stated only in the linear case $p=2$. As a part of the discussion, we provide a few related open problems.

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arXiv2026

On Pleijel-type nodal domain bounds for the $p$-Laplacian

We provide an upper estimate \`a la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{\lambda_k(p;\Omega)\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k \to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality $\lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p)$ in an $N$-dimensional ball $B$, where $\lambda_\ominus(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.

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arXiv2026

On the Hersch-Weinberger inequality in higher dimensions

We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $\Omega \subset \mathbb{R}^N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the first eigenvalue over a class of such domains under perimeter and volume constraints, and under an additional convexity assumption when $N \geq 3$. This result generalizes to a wider class, and extends to higher dimensions, the inequality of Hersch [20], whose approach was substantially based on a construction of the so-called effectless cut by Weinberger [35], so that we call it the Hersch-Weinberger inequality. Our method is based on the analysis of the gradient flow of the first eigenfunction and several approximation procedures, without relying on the effectless cut itself. The effectless cut being a complicated object related to the attractor of the gradient flow, we describe its most fundamental topological properties. In particular, we show that it does not necessarily have to be a hypersurface.

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arXiv2026

Non-Ljusternik--Schnirelman eigenvalues of the pure $p$-Laplacian exist

An old and well-known open problem in the critical point theory asks whether, for some $p \neq 2$ and some bounded domain $\Omega$, there exists a critical value of the $p$-Dirichlet energy $\|\nabla u\|_p^p$ over an $L^p(\Omega)$-sphere in $W_0^{1,p}(\Omega)$ lying outside of a Ljusternik--Schnirelman type sequence of critical values, the latter will be called LS eigenvalues of the $p$-Laplacian. In this work, we provide a positive answer by showing the existence of a non-LS eigenvalue when $p>2$ is sufficiently close to $2$ and $\Omega$ is just a planar rectangle close to the square. The arguments pursue the observation that a simple eigenvalue of the Laplacian can be a meeting point for several branches of eigenvalues of the $p$-Laplacian as $p$ varies. Since LS eigenvalues are continuous with respect to $p$ and exhaust the whole spectrum when $p=2$, we deduce that at least one of the branches must contain non-LS eigenvalues.

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arXiv2026

Symmetry of fractional Neumann eigenfunctions in the ball

We investigate symmetry properties of the first nontrivial eigenfunctions of the fractional Laplacian $(-\Delta)^s$, where $s \in (0,1)$, in an $N$-dimensional ball with nonlocal Neumann boundary conditions. By means of a spectral stability result, we prove that, when $s$ is sufficiently close to $1$, the eigenspace associated to the first nontrivial eigenvalue is generated by $N$ antisymmetric eigenfunctions with exactly two nodal domains in the ball.

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arXiv2026

On the nodal set conjecture for the $p$-Laplacian in circularly symmetric domains

In 1990, P\"utter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain $\Omega$ intersects the boundary $\partial\Omega$ provided $\Omega$ has the circular symmetry. By adopting the method of moving polarization, we establish similar information on the nodal set of second eigenfunctions of the Dirichlet $p$-Laplacian on circularly symmetric domains in arbitrary higher dimension.

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arXiv2025

On Rayleigh quotients connected to $p$-Laplace equations with polynomial nonlinearities

Let $\Omega$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\Omega)$-solutions of the equation $-\Delta_p u = \mu |u|^{q-2}u + |u|^{r-2}u$ parameterized by $\mu$ are in bijection with properly normalized critical points of the $0$-homogeneous Rayleigh type quotient $R_\alpha(u)=\|\nabla u\|_p^p/ (\|u\|_q^{\alpha p} \|u\|_r^{p-\alpha p})$ parameterized by $\alpha$. We study this bijection and properties of $R_\alpha$ for various relations between $p,q,r$. In particular, for the generalized convex-concave problem (the case $q<p<r$) the bijection allows to provide the existence and characterization of all degenerate solutions corresponding to the inflection point of the fibred energy functional: they are critical points of $R_\alpha$ exclusively with $\alpha = (r-p)/(r-q)$. In the subhomogeneous case $q<r \leq p$ and under additional assumptions on $\Omega$, the ground state level of $R_\alpha$ is simple and isolated, and minimizers of $R_\alpha$ exhaust the whole set of sign-constant solutions of the corresponding equation. In the superhomogeneous case $p < q<r$, there are no sign-changing critical points in a vicinity of the ground state level of $R_\alpha$.

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arXiv2025

Reverse Faber-Krahn inequality for planar doubly connected domains

We prove that among all doubly connected and elastically supported planar membranes $Ω$ with prescribed values of the area $|Ω|$ and the lengths of the inner and outer boundaries $|\partial Ω_{\rm{in}}|_1$, $|\partial Ω_{\rm{out}}|_1$ satisfying $|\partial Ω_{\rm{out}}|_1^2 - |\partial Ω_{\rm{in}}|_1^2 = 4π|Ω|$, the concentric annular membrane has the maximal fundamental frequency. The elastic constants $h_{\rm{in}}$, $h_{\rm{out}}$ on $\partial Ω_{\rm{in}}$, $\partial Ω_{\rm{out}}$, respectively, are assumed to satisfy $h_{\rm{in}} \cdot h_{\rm{out}} \geq 0$ and can admit negative values and $+\infty$, the latter being understood as a fixation of the membrane on the corresponding part of the boundary. Our study extends and unifies several existing results in the literature. The case $h_{\rm{in}} \cdot h_{\rm{out}} = 0$ is proved using the method of interior parallels à la Payne & Weinberger, and it requires less restrictive assumptions on $Ω$. For the case $h_{\rm{in}} \cdot h_{\rm{out}} > 0$, we develop the construction of the so-called ``effectless cut'' of $Ω$ described in terms of the gradient flow of the first eigenfunction. This concept was originally introduced by Weinberger and used by Hersch in the fixed boundary case, whose arguments we also revise.

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arXiv2025

Multiplicity of dead core solutions in indefinite elliptic problems

We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$ -Δ_p u = (a^+(x) - μa^-(x))|u|^{q-2}u, \quad u \in W_0^{1,p}(Ω), $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$, $1 0$. Thanks to a dead core formation property, we obtain multiple nonnegative dead core solutions of this problem for $μ$ large enough. This assertion, in combination with a uniqueness feature, yields an exact multiplicity result, which gives a precise description of the nonnegative solutions set of this problem for large values of $μ$. We also extend the multiplicity result to other classes of problems.

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arXiv2025

Inverse iteration method for higher eigenvalues of the $p$-Laplacian

We propose a characterization of a $p$-Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive $p$-Poisson equations. The approach relies on the second eigenvalue's minimax properties, but the actual limiting eigenvalue depends on the choice of initial function. The well-posedness and convergence of the iterative scheme are proved. Moreover, we provide the corresponding numerical computations. As auxiliary results, which also have an independent interest, we provide several properties of certain $p$-Poisson problems.

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arXiv2025

A posteriori estimates for problems with monotone operators

We propose a method of obtaining a posteriori estimates which does not use the duality theory and which applies to variational inequalities with monotone operators, without assuming the potentiality of operators. The effectiveness of the method is demonstrated on problems driven by nonlinear operators of the $p$-Laplacian type, including the anisotropic $p$-Laplacian, polyharmonic $p$-Laplacian, and fractional $p$-Laplacian.

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arXiv2024

Neumann domains of planar analytic eigenfunctions

Along with the partition of a planar bounded domain $\Omega$ by the nodal set of a fixed eigenfunction of the Laplace operator in $\Omega$, one can consider another natural partition of $\Omega$ by, roughly speaking, gradient flow lines of a special type (separatrices) of this eigenfunction. Elements of such partition are called Neumann domains and their boundaries are Neumann lines. When the eigenfunction is a Morse function, this partition corresponds to the Morse--Smale complex and its fundamental properties have been systematically investigated by Band & Fajman (2016). Although, in the case of general position, eigenfunctions are always of the Morse type, particular eigenfunctions can possess degenerate critical points. In the present work, we propose a way to characterize Neumann domains and lines of an arbitrary eigenfunction. Instead of requiring the nondegeneracy of critical points of the eigenfunction, its real analyticity is principally used. The analyticity allows for the presence of degenerate critical points but significantly limits their possible diversity. Even so, the eigenfunction can possess curves of critical points, which have to belong naturally to the Neumann lines set, as well as critical points of a saddle-node type. We overview all possible types of degenerate critical points in the eigenfunction's critical set and provide a numerically based evidence that each of them can be observed for particular eigenfunctions. Alongside with [Band & Fajman, 2016], our approach is inspired by a little-known note of Weinberger that appeared back in 1963, where a part of the Neumann line set, under the name of "effectless cut", was explicitly introduced and studied for the first eigenfunctions in domains with nontrivial topology. In addition, we provide an asymptotic counting of Neumann domains for a disk and rectangles in analogy with the Pleijel constant.

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arXiv2024

Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains

Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-\Delta)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\Omega$ and under the nonlocal zero Dirichlet conditions. Assuming only that $\Omega$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partial\Omega$. As a consequence, the nodal set of $u$ has the same property whenever $\Omega$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.

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arXiv2023

Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions

Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus \overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial \Omega_{\text{out}}$ and the Robin boundary condition with parameter $h \in (-\infty,+\infty]$ on the sphere $\partial B_\alpha$ of radius $\alpha>0$ centered at the origin, the limiting case $h=+\infty$ being understood as the Dirichlet boundary condition on $\partial B_\alpha$. In the case $h>0$, it is known that the first eigenvalue $\tau_1(\Omega)$ does not exceed $\tau_1(B_\beta \setminus \overline{B_\alpha})$, where $\beta>0$ is chosen such that $|\Omega| = |B_\beta \setminus \overline{B_\alpha}|$, which can be regarded as a reverse Faber-Krahn type inequality. We establish this result for any $h \in (-\infty,+\infty]$. Moreover, we provide related estimates for higher eigenvalues under additional geometric assumptions on $\Omega$, which can be seen as Szeg\H{o}-Weinberger type inequalities. A few counterexamples to the obtained inequalities for domains violating imposed geometric assumptions are given. As auxiliary information, we investigate shapes of eigenfunctions associated with several eigenvalues $\tau_{i}(B_\beta \setminus \overline{B_\alpha})$ and show that they are nonradial at least for all positive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. At the same time, we give numerical evidence that, in the planar case $N=2$, already second eigenfunctions can be radial for some $h<0$. The latter fact provides a simple counterexample to the Payne nodal line conjecture in the case of the mixed boundary conditions.

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arXiv2023

Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters

We investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $\alpha,\beta \in \mathbb{R}$. We determine three generally different ranges of parameters $\alpha$ and $\beta$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.

math.AP↗

arXiv2023

Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

We show that the parabolic equation $u_t + (-\Delta)^s u = q(x) |u|^{\alpha-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times (\mathbb{R}^N \setminus \Omega)$, where $\Omega$ is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided $\alpha \in (0,1)$ and the nonnegative bounded weight function $q$ is separated from zero on an open subset of $\Omega$. This fact contrasts with the (super)linear case $\alpha \geq 1$ in which the only bounded finite energy solution is identically zero.

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arXiv2022

Improved Friedrichs inequality for a subhomogeneous embedding

For a smooth bounded domain $\Omega$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - \lambda_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(\Omega)$, where $\lambda_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-\Delta_p u = \lambda_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $\lambda_1$.

math.AP↗

arXiv2022

On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-\Delta_p u = \lambda m(x)|u|^{p-2}u + \eta a(x)|u|^{q-2}u + f(x)$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $q 1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^\gamma(\Omega)$ with $\gamma>N$. For completeness, we also investigate the existence of solutions.

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