We study the exterior and interior Bernoulli problems for half Laplacian problem spectral Laplacian. concentrate on existence geometric properties of solutions. Our main results are following. For Laplacian, we show that under starshapedness assumptions data free domain is starshaped. convexity convex prove a Brunn–Minkowski inequality constant. use variational methods in combination with regul...

Let d be a positive integer and U ⊂ ℤd finite. We study $$\beta (U): = \mathop {\inf }\limits_{\mathop {A,B \ne \phi }\limits_{{\rm{finite}}} } {{\left| {A + B U} \right|} \over {{{\left| A \right|}^{1/2}}{{\left| \right|}^{1/2}}}},$$ other related quantities. employ tensorization, which is not available for the doubling constant, ∣U U∣/∣U∣. For instance, we show (U) \left| \right|,$$ whenever ...

In this paper we define an addition operation on the class of quasiconcave functions. While the new operation is similar to the well-known supconvolution, it has the property that it polarizes the Lebesgue integral. This allows us to define mixed integrals, which are the functional analogs of the classic mixed volumes. We extend various classic inequalities, such as the Brunn-Minkowski and the ...