The stochastic optimal control uses the differential equation of Bellman and its solution - the Bellman function. Recently the Bellman function proved to be an efficient tool for solving some (sometimes old) problems in harmonic analysis.
We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the formOur setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in and and of the Bernstein problem on the flatness of minimal area graphs in . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...
We prove an algebra property under pointwise multiplication for Besov spaces defined on Lie groups of polynomial growth. When the setting is restricted to H-type groups, this algebra property is generalized to paraproduct estimates.
We investigate the behavior of weak solutions to the transmission problem for the Laplace operator with N different media in a neighborhood of a boundary conical point. We establish a precise exponent of the decreasing rate of the solution.