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摘要
We study a Bernoulli type free boundary problem with two phases J[u] = integral(Omega) vertical bar del u(x)vertical bar(2) dx + Phi(M-(u), M+(u)), u - (u) over bar W-0(1,2) (Omega), where (u) over bar is an element of W-1,W-2(Omega) is a given boundary datum. Here, M-1 and M-2 are weighted volumes of {u <= 0} boolean AND Omega and {u > 0} boolean AND Omega, respectively, and Phi is a nonnegative function of two real variables. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in Omega is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. As a special case, we can deal with the energy levels generated by the volume term Phi(0,r(2)) similar or equal to r(2)(n-1/n) , which interpolates the Athanasopoulos-Caffarelli-Kenig-Salsa energy, thanks to the isoperimetric inequality. In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.
- 出版日期2017-12-1
