Regularity of the free boundary for two-phase problems governed by divergence form equations and applications (Q272728)

In this paper the authors study a class of two-phase inhomogeneous free boundary problems governed by elliptic equations in divergence form and prove that Lipschitz or flat free boundaries are \(C^{1,\gamma}\). Their results also apply to the classical Prandtl-Batchelor model in hydrodynamics.NEWLINENEWLINEThis work continues the program developed by the authors in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 121, 382--402 (2015; Zbl 1348.35330)], where the authors construct, via Perron's method, a Lipschitz viscosity solution to problems governed by elliptic equations in divergence form with Hölder continuous coefficients. Weak measure theoretical regularity properties are also proved, such as flatness of the free boundary in a neighborhood of each point of its reduced part. In this paper the authors prove that flat or Lipschitz free boundaries are locally \(C^{1,\gamma}\).NEWLINENEWLINETo describe in more detail the results proved, let \(\Omega\subset\mathbb R^n\) be a bounded, Lipschitz domain and \(A\) a symmetric matrix such that \(A\in C^{0,\overline{\gamma}}(\Omega)\) and it is uniformly elliptic. Denote \(\mathcal L:=\operatorname{div}(A(x)\nabla\cdot)\) and let \(f\in L^\infty(\Omega)\).NEWLINENEWLINEThe problem considered is the two-phase inhomogeneous free boundary problem NEWLINE\[NEWLINE\mathcal Lu=f\text{ in }\Omega^+(u):=\{u>0\}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\mathcal Lu= f \text{ in }\Omega^-(u):=\{u\leq 0\}^\circNEWLINE\]NEWLINE NEWLINE\[NEWLINE|\langle A\nabla u^+,\nabla u^+\rangle|^2 - |\langle A\nabla u^-,\nabla u^-\rangle|^2=1 \text{ on }F(u):=\partial\{u>0\}\cap\Omega.NEWLINE\]NEWLINENEWLINENEWLINEThe main result of this paper is a ``flatness implies regularity'' result: if \(u\) is a viscosity solution to the problem above in \(B_1\), there exists a universal constant \(\overline{\delta}>0\) such that, if NEWLINE\[NEWLINE\{x_n\leq -\delta\}\subset B_1\cap \{u^+(x)=0\}\subset \{x_n \leq\delta\}, NEWLINE\]NEWLINE with \(0\leq\delta\leq\overline{\delta}\), then \(F(u)\) is \(C^{1,\gamma}\) in \(B_{1/2}\) for some universal \(\gamma\in (0,1)\).NEWLINENEWLINEThe strategy of the proof of this theorem follows the one in [the authors, Anal. PDE 7, No. 2, 267--310 (2014; Zbl 1296.35218)].NEWLINENEWLINEThe key tools are a Harnack type inequality and an improvement of flatness lemma which allows to linearize the problem into a standard transmission problem.NEWLINENEWLINEThe second main result of this paper is a ``Lipschitz implies regularity'' result: if \(u\) is a viscosity solution to the problem above in \(B_1\) and \(F(u)\) is Lipschitz in \(B_1\), then \(F(u)\) is \(C^{1,\gamma}\) in \(B_{1/2}\) for some universal \(\gamma\in (0,1)\).NEWLINENEWLINEThis theorem follows from the flatness result via a blow-up argument and the regularity result for the homogeneous problem with \(A\equiv I\). To prove this regularity result the authors use a Weiss type monotoni\-city formula, which together with the Alt-Caffarelli-Friedman monotonicity formula provides a new proof of the regularity result for the homogeneous problem with the Laplace operator.NEWLINENEWLINEAs a consequence of the flatness theorem, a regularity result for the minimal Perron solution \(u\) follows. More precisely, it is proved that if \(u\) is the Perron solution, in a neighborhood of every \(x_0\in F^*(u)\) (which denotes the reduced free boundary), \(F(u)\) is a \(C^{1,\gamma}\) surface.NEWLINENEWLINEFinally, the authors apply their results to the classical Prandtl-Batchelor model in hydrodynamics.NEWLINENEWLINEConsider a bounded domain \(\Omega\subset \mathbb R^2\) and \(\mu, w>0\). One looks for a function \(u\) with \(u=\mu\) on \(\partial\Omega\) such that NEWLINE\[NEWLINE\Delta u=0\text{ in }\Omega^+(u), \quad \Delta u=w \text{ in } \Omega^-(u) NEWLINE\]NEWLINE NEWLINE\[NEWLINE|\nabla u^+|^2-|\nabla u^-|^2=\sigma \text{ on } F(u):=\Omega\cap\partial\Omega^+(u), NEWLINE\]NEWLINE where \(\sigma>0\).NEWLINENEWLINEGiven an appropriate condition, the authors prove existence of a Perron solution \(u\) of the Prandtl-Batchelor problem. In particular, the free boundary \(F(u)\) has \(\mathcal H^1\) finite measure and in a neighborhood of any point of the reduced boundary, \(F^\ast(u)\) is a \(C^{1,\gamma}\) curve.