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Vladimir Bobkov

Publications and source records attributed to Vladimir Bobkov.

At least 37 records · Page 2Linked to original sources
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 $-Δ_p u = λm(x)|u|^{p-2}u + ηa(x)|u|^{q-2}u + f(x)$ in a bounded domain $Ω\subset \mathbb{R}^N$, 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 $m,f \in L^γ(Ω)$ with $γ>N$. For completeness, we also investigate the existence of solutions.

math.AP↗
arXiv2022

Basisness and completeness of Fucik eigenfunctions for the Neumann Laplacian

We investigate the basis properties of sequences of Fucik eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in $L^2(0,π)$ and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fucik eigenvalues which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,π)$ and we explicitly describe the corresponding biorthogonal system.

math.CA↗
arXiv2021

Basisness of Fucik eigenfunctions for the Dirichlet Laplacian

We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,π)$. For that purpose, we introduce a criterion for a sequence in a Hilbert space to be a Riesz basis.

math.CA↗
arXiv2021

Szegő-Weinberger type inequalities for symmetric domains with holes

Let $μ_2(Ω)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $Ω\subset\mathbb{R}^N$. It was proved by Szegő for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains $μ_2(Ω)$ attains its global maximum if $Ω$ is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szegő-Weinberger result for a class of domains of the form $Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}$ which are either centrally symmetric or symmetric of order $2$ (with respect to every coordinate plane $(x_i,x_j)$) by showing that $μ_{2}(Ω_{\text{out}}\setminus\overlineΩ_{\text{in}})\leqμ_2(B_β\setminus\overline{B}_α)$, where $B_α, B_β$ are balls centered at the origin such that $B_α\subsetΩ_{\text{in}}$ and $|Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}|=|B_β\setminus\overline{B}_α|$. Secondly, we provide Szegő-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if $Ω_{\text{out}}\setminus\overlineΩ_{\text{in}}$ is symmetric of order $4$, then we prove $μ_{i}(Ω_{\text{out}}\setminus\overlineΩ_{\text{in}})\leqμ_i(B_β\setminus\overline{B}_α)$ for $i=3,\dots,N+2$, where we also allow $Ω_{\text{in}}$ and $B_α$ to be empty. If $N=2$ and the domain is symmetric of order $8$, then the latter inequality persists for $i=5$. Counterexamples to the obtained inequalities for domains outside of the considered symmetry classes are given. The existence and properties of nonradial domains with required symmetries in higher dimensions are discussed. As an auxiliary result, we obtain the non-radiality of the eigenfunctions associated to $μ_{N+2}(B_β\setminus\overline{B}_α)$.

math.AP↗
arXiv2021

On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-Δ_p u = λ|u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $1 0\}$, when the parameter $λ$ lies in a neighborhood of the critical value $λ^* = \inf\left\{\int_Ω|\nabla u|^p \, dx/\int_Ω|u|^p \, dx: u\in W_0^{1,p}(Ω) \setminus \{0\},\ \int_Ωa|u|^q\,dx \geq 0\,\right\}$. Among main results, we show that if $p>2q$ and either $\int_Ωaφ_p^q\,dx=0$ or $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $λ^*$. Here $φ_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $Ω$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q 2q$ and $\int_Ωaφ_p^q\,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $λ^*$, two of which are strictly positive in $\{x\in Ω: a(x)>0\}$.

math.AP↗
arXiv2021

Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters

We study the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = α|u|^{p-2}u+β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(α,β)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(α,β) = \left(\|\nabla φ_p\|_p^p/\|φ_p\|_p^p, \|\nabla φ_p\|_q^q/\|φ_p\|_q^q\right)$, where $φ_p$ is the first eigenfunction of the $p$-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents $p$ and $q$.

math.AP↗
arXiv2021

Sensitivity of Microwave Interferometer in the Limiter Shadow to filaments in ASDEX Upgrade

Microwave interferometer in the Limiter Shadow (MILS) is a new diagnostic, installed on ASDEX Upgrade for electron density measurements in the far Scrape-Off Layer (SOL). At the chosen frequency of 47 GHz the region of measurements varies within several centimeters before and after the limiter, depending on the density. 200 kHz data acquisition allows resolving transient events such as edge localised modes (ELMs) filaments and turbulence filaments. The measured quantities, phase shift and power decay of the microwave beam, which crosses the plasma, are directly connected to the density and do not depend on any other plasma quantity. In this work, we analyse the influence of a filamentary perturbation on MILS signals. Simple representation of a filament is adopted, with parameters relevant to experimental filament properties, reported for ASDEX Upgrade. Forward modelling is done in COMSOL software by using RAPLICASOL, to study the response of the MILS synthetic diagnostic to the presence of a filament. Qualitative and quantitative dependencies are obtained and the boundaries of MILS sensitivity to filaments, or to the density perturbation in far SOL in general, are outlined.

physics.plasm-ph↗
arXiv2021

On the Cheeger problem for rotationally invariant domains

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Ω\subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap Ω$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $Ω$ is convex, then the free boundary of $C$ consists only of pieces of spheres and nodoids. This result remains valid for nonconvex domains when the generating curve of $C$ is closed, convex, and of class $\mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the fact that, for general nonconvex domains, pieces of unduloids or cylinders can also appear in the free boundary of $C$.

math.OC↗
arXiv2021

Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = f_μ(x,u,\nabla u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_μ(x,u,\nabla u) = λ_1(p) |u|^{p-2} u + μ|u|^{q-2} u$, where $λ_1(p)$ is the first eigenvalue of the $p$-Laplacian.

math.AP↗
arXiv2020

Basis properties of Fucik eigenfunctions

We establish sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,π)$.

math.CA↗
arXiv2020

Estimates on the spectral interval of validity of the anti-maximum principle

The anti-maximum principle for the homogeneous Dirichlet problem to $-Δ_p u = λ|u|^{p-2}u + f(x)$ with positive $f \in L^\infty(Ω)$ states the existence of a critical value $λ_f > λ_1$ such that any solution of this problem with $λ\in (λ_1, λ_f)$ is strictly negative. In this paper, we give a variational upper bound for $λ_f$ and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem in $(λ_1,λ_2)$.

math.AP↗
arXiv2020

Existence and multiplicity results for a class of semilinear elliptic equations

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Δu = (1-u) u^m - λu^n$ in a bounded domain $Ω\subset \mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0 0$. When $λ> 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.

math.AP↗
arXiv2020

On asymptotic behaviour of Dirichlet inverse

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.

math.NT↗
arXiv2019

On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem $-Δ_p u = f(u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ upon domain perturbations. Assuming that the nonlinearity $f$ is superlinear and subcritical, we establish Hadamard-type formulas for such critical levels. As an application, we show that among all (generally eccentric) spherical annuli $Ω$ least nontrivial critical levels attain maximum if and only if $Ω$ is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions whenever $Ω$ is a ball or concentric annulus.

math.AP↗
arXiv2019

Second-order derivative of domain-dependent functionals along Nehari manifold trajectories

Assume that a family of domain-dependent functionals $E_{Ω_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers of $E_{Ω_t}$ over the associated Nehari manifold $\mathcal{N}(Ω_t)$. We obtain a formula for the second-order derivative of $E_{Ω_t}$ with respect to $t$ along Nehari manifold trajectories of the form $α_t(u_0(Φ_t^{-1}(y)) + t v (Φ_t^{-1}(y)))$, $y \in Ω_t$, where $Φ_t$ is a diffeomorphism such that $Φ_t(Ω_0) = Ω_t$, $α_t \in \mathbb{R}$ is a $\mathcal{N}(Ω_t)$-normalization coefficient, and $v$ is a corrector function whose choice is fairly general. Since $E_{Ω_t}[u_t]$ is not necessarily twice differentiable with respect to $t$ due to the possible nonuniqueness of $u_t$, the obtained formula represents an upper bound for the corresponding second superdifferential, thereby providing a convenient way to study various domain optimization problems related to $E_{Ω_t}$. An analogous formula is also obtained for the first eigenvalue of the $p$-Laplacian. As an application of our results, we investigate the behaviour of the first eigenvalue of the Laplacian with respect to particular perturbations of rectangles.

math.AP↗
arXiv2019

On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities

We show that the elliptic equation with a non-Lipschitz right-hand side, $-Δu = λ|u|^{β-1}u - |u|^{α-1}u$ with $λ>0$ and $0<α<β<1$, considered on a smooth star-shaped domain $Ω$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $Γ$ of the boundary $\partialΩ$ such that $Γ\neq \partialΩ$.

math.AP↗
arXiv2019

On the Fredholm-type theorems and sign properties of solutions for $(p,q)$-Laplace equations with two parameters

We consider the Dirichlet problem for the nonhomogeneous equation $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u + f(x)$ in a bounded domain, where $p \neq q$, and $α, β\in \mathbb{R}$ are parameters. We explore assumptions on $α$ and $β$ that guarantee the resolvability of the considered problem. Moreover, we introduce several curves on the $(α,β)$-plane allocating sets of parameters for which the problem has or does not have positive or sign-changing solutions, provided $f$ is of a constant sign.

math.AP↗