arXiv subjects

Sergey Kolonitskii

Publications and source records attributed to Sergey Kolonitskii.

4 recordsLinked to original sources

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 $(-Δ)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $Ω$ and under the nonlocal zero Dirichlet conditions. Assuming only that $Ω$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partialΩ$. As a consequence, the nodal set of $u$ has the same property whenever $Ω$ 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.

math.AP↗

arXiv2022

Improved Friedrichs inequality for a subhomogeneous embedding

For a smooth bounded domain $Ω$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - λ_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(Ω)$, where $λ_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-Δ_p u = λ_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $λ_1$.

math.AP↗

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↗