We consider the problemwhere $\Omega \subset {ℝ}^3$ is a smooth and bounded domain, $\epsilon ,{\gamma }_1,{\gamma }_2\>0,$ $v,V:\Omega \to ℝ$, $f:ℝ\to ℝ$. We prove that this system has a $v_{\epsilon }$ which develops, as $\epsilon \to 0^{+}$, a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches thepart of $\partial \Omega$,, where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem...

For a bounded and sufficiently smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $N\ge 2$, let ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$ and ${\left({\phi }_k\right)}_{k=1}^{\mathrm{\infty }}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$-\text{div}\left(a\left(x\right)\nabla {\phi }_k\right)+q\left(x\right){\phi }_k={\lambda }_k\varrho \left(x\right){\phi }_k\text{ in }\mathrm{\Omega },a\frac{\partial }{\partial \mathbf{n}}{\phi }_k=0\text{ su }\partial \mathrm{\Omega }.$$ We prove that knowledge of the Dirichlet boundary spectral data ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$, ${\left({\phi }_{k|\partial \mathrm{\Omega }}\right)}_{k=1}^{\mathrm{\infty }}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a,q,\varrho$ their identifiability is then proved. We prove also analogous results for...

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem: $$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$ where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$ ${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D=B_1^{+}\left(0\right),$ then we impose in addition that $$u\left(x\right)\equiv 0\text{on}\left\{(x^{\text{'}},x_n)\right|x_n=0\}.$$ We show that a fairly mild thickness assumption on ${\Omega }^c$ will ensure enough compactness on...

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in ${ℝ}^4$, and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which...

We address the numerical minimization of the functional $\mathcal{F}\left(v\right)={\int }_{\mathrm{\Omega }}\left|Dv\right|+{\int }_{\partial \mathrm{\Omega }}\mu vd{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}xvdx$, for $v\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$. We note that $\mathcal{F}$ can be equivalently minimized on the larger, convex, set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and that, on that space, $\mathcal{F}$ may be regularized with a sequence $\mathit{\left\{}\mathcal{F}{}_{ϵ}\right(v)=\int {}_{\mathrm{\Omega }}\sqrt{{ϵ}^2+{\left|Dv\right|}^2}+\int {}_{\partial \mathrm{\Omega }}\mu vd\mathcal{H}{}^{n-1}-\int {}_{\mathrm{\Omega }}xvdx\mathit{\}}{}_{ϵ}$of regular functionals. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$ can be discretized by continuous linear finite elements. The convexity of the functionals in $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful for the numerical minimization of $\mathcal{F}$. We prove the $\mathrm{\Gamma }-L^1\left(\mathrm{\Omega }\right)$-convergence of the discrete functionals to $\mathcal{F}$ and present a few numerical examples. ...