Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface. (Q2856921)

Let \(\Omega \subset \mathbb {R}^3\) be a bounded domain with boundary \(\Gamma \) of class \(C^2\). Let \(S\subset \Omega \) be a bounded connected surface of class \(C^2\) with boundary such that \(S\cap \Gamma \) is a closed curve and \(\Omega \setminus S=\Omega _1\cup \Omega _2\), where \(\Omega _1\) and \(\Omega _2\) are two disjoint open sets. Finally, let \(n_\Gamma \) be the outward unit normal to \(\Gamma \) and \(n_S\) the unit normal to \(S\) that points to \(\Omega _2\).NEWLINENEWLINEThe author studies the Lipschitz continuity of the second derivatives of the weak solutions to the following elliptic transmission problems: NEWLINE\[NEWLINE -\text{div} (\kappa \nabla u)= f \;\text{ in } \Omega \setminus S, NEWLINE\]NEWLINE NEWLINE\[NEWLINE [u]_S=0, \quad [-\kappa \nabla u\cdot n_S]_S=0 \;\text{ on } S, NEWLINE\]NEWLINE associated to one of the following boundary conditions on \(\Gamma \): NEWLINE\[NEWLINE -\kappa \nabla u\cdot n_{\Gamma }= Q \;\text{ on } \Gamma , NEWLINE\]NEWLINE NEWLINE\[NEWLINE u=u_e \;\text{ on } \Gamma . NEWLINE\]NEWLINE Here, the functions \(Q\), \(u_e\) are given, the function \(f\) belongs to \(L^q(\Omega )\) (\(q>1\)), and the diffusion coefficient \(\kappa \) is supposed to be phase-dependent, that is, \(\kappa =k_i(x)\) if \(x\in \Omega \), where \(k_i\: \overline {\Omega _i}\rightarrow \mathbb {R}^{3\times 3}\). The symbol \([\cdot ]_S\) denotes the jump of a quantity across \(S\). Let \(\alpha \) be the angle of the contact of the surfaces \(S\) and \(\Gamma \). The author proves that the unique weak solution \(u\in W^{1,2}(\Omega )\) of the above problem has the higher regularity \(u\in W^{2,2}(\Omega _i)\) (\(i=1,2\)) if \(q=2\) and \(u\in W^{1,\infty }(\Omega )\) if \(q>3\), provided that: {\parindent=0.5cm\begin{itemize}\item[--]\(\alpha \in W^{1,\infty }(\Gamma \cap S)\) and \(\alpha (\Gamma \cap S)\subset ]0,\pi [\); \item[--]\(\kappa \) satisfies certain symmetry conditions and uniform ellipticity conditions; \item[--] a compatibility condition for high regularity which involves the angle \(\alpha \) and the coefficient \(\kappa \) is satisfied; \item[--] some further conditions involving the boundary data \(Q\) and \(u_e\) and the exponent \(q\) are also satisfied. NEWLINENEWLINE\end{itemize}} Finally, if \(q=2\) and the compatibility condition is violated, the author also obtains the following regularity for the solution \(u\: \nabla u\in L^{q_0}(\Omega )\) for some \(q_0>3\) and NEWLINE\[NEWLINE \text{ for any } p\in \left [1,\frac {2\min \{q_0,6\}}{\min \{q_0,6\}+2}\right [,\; \nabla u\in W^{1,p}(\Omega _i)\; (i=1,2). NEWLINE\]