We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form$$\mathrm{div}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${ℝ}^2$ and ${ℝ}^3$ and of the Bernstein problem on the flatness of minimal area graphs in ${ℝ}^3$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

The role of the second critical exponent $p=(n+1)/(n-3)$, the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega$ in ${ℝ}^n$ with bounded, smooth boundary. Given $\Gamma$, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3-\epsilon )$, there exists a solution $u_{\epsilon }$ such that $|\nabla u_{\epsilon }{|}^2$ converges weakly to a Dirac measure on $\Gamma$ as $\epsilon \to 0^{+}$, provided that $\Gamma$ is nondegenerate in the sense of second variations of...

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0\text{in}\Omega$, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $d^2{\left|\nabla u_d\right|}^2\rightharpoonup S{\delta }_K\mathrm{as}d\to 0$ in the sense of measures, where ${\delta }_K$...

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^1$-valued function defined on the boundary of a bounded regular domain of $R^n$. When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...