Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

Poincaré Inequality on Subanalytic Sets

Let $$\Omega $$ Ω be a subanalytic connected bounded open subset of $$\mathbb {R}^n$$ R n , with possibly singular boundary. We show that given $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , there is a constant C such that for any $$u\in W^{1,p}(\Omega )$$ u ∈ W 1 , p ( Ω ) we have $$||u-u_{\Omega }||_{L^p} \le C||\nabla u||_{L^p},$$ | | u - u Ω | | L p ≤ C | | ∇ u | | L p , where we have set $$u_{\Omega }:=\frac{1}{|\Omega |}\int _{\Omega } u.$$ u Ω : = 1 | Ω | ∫ Ω u .

Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction

<p style='text-indent:20px;'>In the paper under study, we consider the following coupled non-degenerate Kirchhoff system</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE0"> \begin{document}$\begin{equation} \left \{ \begin{aligned} &amp; y_{tt}-\mathtt{φ}\Big(\int_\Omega | \nabla y |^2\,dx\Big)\Delta y +\mathtt{α} \Delta \mathtt{θ} = 0, &amp;{\rm{ in }}&amp;\; \Omega \times (0, +\infty)\\ &amp; \mathtt{θ}_t-\Delta \mathtt{θ}-\mathtt{β} \Delta y_t = 0, &amp;{\rm{ in }}&amp;\; \Omega \times (0, +\infty)\\ &amp; y = \mathtt{θ} = 0,\; &amp;{\rm{ on }}&amp;\;\partial\Omega\times(0, +\infty)\\ &amp; y(\cdot, 0) = y_0, \; y_t(\cdot, 0) = y_1,\;\mathtt{θ}(\cdot, 0) = \mathtt{θ}_0, \; \; &amp;{\rm{ in }}&amp;\; \Omega\\ \end{aligned} \right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded open subset of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mathtt{α} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathtt{β} $\end{document}</tex-math></inline-formula> be two nonzero real numbers with the same sign and <inline-formula><tex-math id="M5">\begin{document}$ \mathtt{φ} $\end{document}</tex-math></inline-formula> is given by <inline-formula><tex-math id="M6">\begin{document}$ \mathtt{φ}(s) = \mathfrak{m}_0+\mathfrak{m}_1s $\end{document}</tex-math></inline-formula> with some positive constants <inline-formula><tex-math id="M7">\begin{document}$ \mathfrak{m}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mathfrak{m}_1 $\end{document}</tex-math></inline-formula>. So we prove existence of solution and establish its exponential decay. The method used is based on multiplier technique and some integral inequalities due to Haraux and Komornik[<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b8">8</xref>].</p>

Local boundedness for minimizers of convex integral functionals in metric measure spaces

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

Morrey estimates for a class of elliptic equations with drift term

We consider the following boundary value problem $$\begin{array}{} \displaystyle \begin{cases} - {\rm div}{[M(x)\nabla u - E(x) u]} =f(x) & \text{in}~~ {\it\Omega} \\ u =0 & \text{on}~~ \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω is a bounded open subset of ℝN, with N > 2, M : Ω → ℝN2 is a symmetric matrix, E(x) and f(x) are respectively a vector field and function both belonging to suitable Morrey spaces and we study the corresponding regularity of u and D u.

Local regularity results for solutions of linear elliptic equations with drift term

We study the local regularity of the solution u of the following nonlinear boundary value problem: \left\{\begin{aligned} \displaystyle\mathcal{A}u&\displaystyle=-\operatorname{% div}{[E(x)u+F(x)]}&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded open subset of {\mathbb{R}^{N}} , with {N>2} , {\mathcal{A}} is a nonlinear Leray–Lions operator in divergence form, and {E(x)} and {F(x)} are vector fields satisfying suitable local summability properties.

THE DISTRIBUTIONAL -HESSIAN IN FRACTIONAL SOBOLEV SPACES

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

A uniqueness result for some singular semilinear elliptic equations

Given [Formula: see text] a bounded open subset of [Formula: see text], we consider non-negative solutions to the singular semilinear elliptic equation [Formula: see text] in [Formula: see text], under zero Dirichlet boundary conditions. For [Formula: see text] and [Formula: see text], we prove that the solution is unique.

Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

We consider an elastic thin film as a bounded open subset