arXiv subjects

Joao V. da Silva

Publications and source records attributed to Joao V. da Silva.

3 recordsLinked to original sources

arXiv2017

Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty-$eigenvalues

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $Ω$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mathfrak{B}_r)| = δ_2, $$ then there are two balls such that $$\mathfrak{B}_{\frac{r}{δ_1 r+1}} \subset Ω\subset \mathfrak{B}_{\frac{r+δ_2 r}{1-δ_2 r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $\infty-$eigen-functions.

math.AP↗

arXiv2017

Maximal solutions for the Infinity-eigenvalue problem

In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -Δ_\infty v,\, |\nabla v|-λ_{1, \infty}(Ω) v \} & = & 0 & \text{in} & Ωv & = & 0 & \text{on} & \partial Ω, \end{array} \right. $$ has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as $\ell \nearrow 1$ of concave problems of the form $$ \left\{ \begin{array}{rclcl} \min\{ -Δ_\infty v_{\ell},\, |\nabla v_{\ell}|-λ_{1, \infty}(Ω) v_{\ell}^{\ell} \} & = & 0 & \text{in} & Ωv_{\ell} & = & 0 & \text{on} & \partial Ω. \end{array} \right. $$ In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the concave problems as happens for the usual eigenvalue problem for the $p-$Laplacian for a fixed $1<p<\infty$.

math.AP↗

arXiv2017

The infinity-Fucik spectrum

In this article we study the behavior as $p \nearrow+\infty$ of the Fucik spectrum for $p$-Laplace operator with zero Dirichlet boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$. We characterize the limit equation, and we provide a description of the limit spectrum. Furthermore, we show some explicit computations of the spectrum for certain configurations of the domain.

math.AP↗