\(L^ p\)-approach to mixed boundary value problems for second-order elliptic operators
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) with a \(C^ \infty\) boundary. Let \(A_ p\) and \(L_ p\) be linear differential operators with real-valued coefficients in \(C^ \infty (\overline {\Omega})\) and \(C^ \infty (\partial\Omega)\) respectively, defined in \(H^{s,p} (\Omega)\) \((s\in [2,+ \infty)\), \(p\in (1,+ \infty))\) by the equations \[ \begin{aligned} A_ p u &= \sum_{i,j=1}^ N D_ i[ a^{i,j} (x)D_ j u]+ \sum_{j=1}^ N b^ j(x) D_ j u+ c(x)u \qquad (D_ j= \partial/ \partial x_ j),\\ L_ p u &= \biggl[ a(x) {{\partial u} \over {\partial\nu}}+ b(x)u \biggr]_{| \partial \Omega} \qquad \biggl( \nu_ i= \sum_{j=1}^ N a^{i,j} (x) n_ j \biggr). \end{aligned} \] Here \(n(x)\) denotes the outward normal unit vector at \(x\in \partial\Omega\). Moreover, the matrix \((a^{i,j})_{i, j}\) is assumed to be symmetric and positive definite in \(\overline {\Omega}\) and the function \(a\), entering the boundary opertor \(L_ p\), may vanish on \(\partial\Omega\). Consider then the Banach space \(B_{(a,b)}^{ s-1-1/ p,p} (\partial\Omega)\) defined by \[ B_{(a,b)}^{ s-1-1/ p,p} (\partial\Omega)= \{\varphi= a\varphi_ 1+ b\varphi_ 0:\;\varphi_ j\in H^{s-j-1/ p,p} (\partial\Omega),\;j=0,1\} \] equipped with the norm \[ |\varphi |_{a,b; s-1-1/ p,p}= \inf\{| \varphi_ 1 |_{a,b; s-1-1/ p,p}+ | \varphi_ 0 |_{a,b; s-1/ p,p}:\;\varphi= a\varphi_ 1+ b\varphi_ 0\}. \] Under the conditions \(a(x)\geq 0\), \(b(x)\geq 0\) on \(\partial\Omega\), \(b(x)>0\) on \(\Gamma= \{x\in \partial\Omega\): \(a(x)= 0\}\), \(c(x)<0\) in \(\Omega\) the author shows that the elliptic boundary value problem \(A_ p u=f\) in \(\Omega\), \(L_ p u=\varphi\) on \(\partial\Omega\) admits a unique solution \(u\in H^{s,p} (\Omega)\) continuously depending on the data with respect to the norms pointed out. If, moreover, \(b(x)- \sum_{i=1}^ N b^ i(x) n_ i(x) \geq 0\) \(\forall x\in \partial\Omega\), the author extends to the present case the basic spectral results proved by \textit{S. Agmon} (when \(a(x)>0\) \(\forall x\in \partial\Omega)\) in [Commun. Pure Appl. Math. 15, 119-147 (1962; Zbl 0109.327); and in the book ``Lectures on elliptic boundary value problems'' (1965; Zbl 0142.374)].
scientific article; zbMATH DE number 645687