We study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D subset of R^d, Lambda > 0, and phi_i in H^1/2 (partial derivative D), we deal with a vectorial Bernoulli problem in the box D, with boundary data phi_i that can change sign. We prove that, for any optimal vector U = (u(1),...,u(k)), the free boundary partial {|U|>0} cap D is made of a regular part, which is relatively open and locally the graph of a smooth function, a (one-phase) singular part, of Hausdorff dimension at most d - d*, for a d* in{5,6,7} and by a set of branching (two-phase) points, which is relatively closed and of finite H^d-1 measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.
Regularity of the free boundary for the vectorial Bernoulli problem
Mazzoleni, Dario;
2020-01-01
Abstract
We study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure D subset of R^d, Lambda > 0, and phi_i in H^1/2 (partial derivative D), we deal with a vectorial Bernoulli problem in the box D, with boundary data phi_i that can change sign. We prove that, for any optimal vector U = (u(1),...,u(k)), the free boundary partial {|U|>0} cap D is made of a regular part, which is relatively open and locally the graph of a smooth function, a (one-phase) singular part, of Hausdorff dimension at most d - d*, for a d* in{5,6,7} and by a set of branching (two-phase) points, which is relatively closed and of finite H^d-1 measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
