We consider the following eigenvalue optimization problem: Given a bounded domain ⊂ R and numbers α > 0, A ∈ [0, | |], find a subset D ⊂ of area A for which the first Dirichlet eigenvalue of the operator − + αχ D 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 ; on the other hand, for convex reflection symmetries are preserved.Also, we present numerical results and formulate some conjectures suggested by them.

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Abstract. We show a unique continuation theorem for the Schrödinger equation ( 1 i ∇ − A) 2 u + V u = 0 with singular coefficients A and V . Main resultsIn this paper we show a unique continuation theorem for the Schrödinger operator H = (2 + V with singular magnetic field. In fact we shall establish, under some assumptions on A and V , the following estimate:The latter holds for r > 0 and x o ∈ Ω with B 2r

The paper studies the heat kernel of the Schro$ dinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Ho$ lder class which includes nonnegative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.

In this paper, continuing our earlier article [CGIKO], we study qualitative properties of solutions of a certain eigenvalue optimization problem. Especially we focus on the study of the free boundary of our optimal solutions on general domains.