Boundary value problems for second-order elliptic equations and related topics.

Abstract

We study perturbation results for boundary value problems for second-order elliptic partial differential equations, and the exponential decay of solutions to generalized Schr\"odinger operators. First, through the use of sawtooth domains and the extrapolation technique of Carleson measures, we show the stability of the solvability of the Dirichlet problem for (additive) Carleson perturbations of certain degenerate elliptic operators $-\text{div }A\nabla$ on domains with low dimensional boundaries (joint work with S. Mayboroda). Then, with a different method of proof, we expand these perturbation results to more abstract domains (including some domains with mixed-dimensional boundaries) and a broader type of Carleson perturbation, yielding some new applications (including to free boundary problems) (joint work with J. Feneuil). Next, together with S. Bortz, S. Hofmann, J.L. Luna Garc\'ia, and S. Mayboroda, we consider the uniformly elliptic operators $L=-\text{div }(A\nabla+B_1)+B_2\nabla+V$ on the upper half space $\mathbb R^{n+1}_+=\mathbb R^n\times\{t>0\}$, $n\geq3$, with $t-$independent coefficients, and we prove the $L^2$ solvability of the Dirichlet, Neumann and Regularity problems under the condition that $|B_1|,|B_2|,|V|^2$ have small $L^n(\mathbb R^n)$ norm. Finally, we show that for generalized magnetic Schr\"odinger operators $-(\nabla-i{\bf a})^TA(\nabla-i{\bf a})+V$, with certain conditions providing an uncertainty principle, resolvents and Lax-Milgram solutions exhibit exponential decay (in an $L^2-$sense), and we improve these estimates to upper pointwise exponential decay for the magnetic Schr\"odinger operator $-(\nabla-i{\bf a})^2$, and to sharp (that is, upper and lower) pointwise exponential decay for the Schr\"odinger operator on a non-homogeneous medium $-\text{div }A\nabla +V$ (joint work with S. Mayboroda).