Sagun Chanillo

Rotational Symmetry of Solutions of Some Nonlinear Problems in Statistical Mechanics and in Geometry

The method of moving planes is used to establish a weak set of conditions under which the nonlinear equation −Δu(x)=V(|x|)eu(x),x∈ℝ2 admits only rotationally symmetric solutions. Additional structural invariance properties of the equation then yield another set of conditions which are not originally covered by the moving plane technique. For instance, nonmonotonicV can be accommodated. Results for −Δu(y)=V(y)eu(y)−c, withy∈S2, are obtained as well. A nontrivial example of broken symmetry is also constructed. These equations arise in the context of extremization problems, but no extremization arguments are employed. This is of some interest in cases where the extremizing problem is neither manifestly convex nor monotone under symmetric decreasing rearrangements. The results answer in part some conjectures raised in the literature. Applications to logarithmically interacting particle systems and geometry are emphasized.

Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes

Abstract: We consider the following eigenvalue optimization problem: Given a bounded domain Ω⊂ℝ and numbers α > 0, A∈[ 0, |Ω|], find a subset D⊂Ω of area A for which the first Dirichlet eigenvalue of the operator −Δ+αχD is as small as possible.We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than Ω on the other hand, for convex Ω reflection symmetries are preserved.Also, we present numerical results and formulate some conjectures suggested by them.

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.

Unique continuation for Δ+ and the C. Fefferman-Phong class

We show that the strong unique continuation property holds for the inequality |Δu|≤|v||u|, where the potential v(x) satisfies the C. Fefferman-Phong condition in a certain range of p values. We also deal with the situation of u(x) vanishing at infinity. These are all consequences of appropriate Carleman inequalities

Weak type estimates for Cesàro sums of Jacobi polynomial series

Facts and definitions An absolute value estimate for

Embeddability for 3-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants

3(1-y)\leq 2(1-x)

Surfaces with prescribed Gauss curvature

A basic estimate for

Weighted weak (1,1) and weighted ^{} estimates for oscillating kernels

3(1-y)\leq 2(1-x)

On diameters of uniformly rotating stars

A kernel estimate for

Sharp function and weighted Lp estimates for a class of pseudo-differential operators

3(1-y)\leq 2(1-x)