An introduction to the study of critical points of solutions of elliptic and parabolic equations

We give a survey at an introductory level of old and recent results in the study of critical points of solutions of elliptic and parabolic partial differential equations. To keep the presentation simple, we mainly consider four exemplary boundary value problems: the Dirichlet problem for the Laplace's equation; the torsional creep problem; the case of Dirichlet eigenfunctions for the Laplace's equation; the initial-boundary value problem for the heat equation. We shall mostly address three issues: the estimation of the local size of the critical set; the dependence of the number of critical points on the boundary values and the geometry of the domain; the location of critical points in the domain.Comment: 34 pages, 13 figures; a few slight changes and some references added; to appear in the special issue, in honor of G. Alessandrini's 60th birthday, of the Rendiconti dell'Istituto Matematico dell'Universit\`a di Triest

The Matzoh Ball Soup Problem: a complete characterization

We characterize all the solutions of the heat equation that have their (spatial) equipotential surfaces which do not vary with the time. Such solutions are either isoparametric or split in space-time. The result gives a final answer to a problem raised by M. S. Klamkin, extended by G. Alessandrini, and that was named the Matzoh Ball Soup Problem by L. Zalcman. Similar results can also be drawn for a class of quasi-linear parabolic partial differential equations with coefficients which are homogeneous functions of the gradient variable. This class contains the (isotropic or anisotropic) evolution p-Laplace and normalized p-Laplace equations

Interaction between nonlinear diffusion and geometry of domain

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider nonlinear diffusion equations of the form $\partial_t u= \Delta \phi(u)$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $\Omega$, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^N\setminus \Omega$. We consider an open ball $B$ in $\Omega$ whose closure intersects $\partial\Omega$ only at one point, and we derive asymptotic estimates for the content of substance in $B$ for short times in terms of geometry of $\Omega$. Also, we obtain a characterization of the hyperplane involving a stationary level surface of $u$ by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.Comment: 25 pages, no figures. Added some details to introduction. A couple of small changes. To appear in Journal Diff. Eq

A note on Serrin's overdetermined problem

We consider the solution of the torsion problem $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial \Omega$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_\nu$ is constant on $\partial \Omega$, then $\Omega$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$r_e-r_i\leq C_t\,\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partial\Omega$ and $\Gamma_t$ is a surface parallel to $\partial\Omega$ at distance $t$ and sufficiently close to $\partial\Omega$; secondly, if in addition $u_\nu$ is constant on $\partial\Omega$, show that \max_{\Gamma_t} u-\min_{\Gamma_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains $\Omega$ are ellipses

Analytical results for 2-D non-rectilinear waveguides based on the Green's function

We consider the problem of wave propagation for a 2-D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case.Comment: 18 pages, 8 figure