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Bilinear embedding for divergence-form operators with complex coefficients on irregular domains
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-05-21 , DOI: 10.1007/s00526-020-01751-3
Andrea Carbonaro , Oliver Dragičević

Let \(\Omega \subseteq {\mathbb {R}}^{d}\) be open and A a complex uniformly strictly accretive \(d\times d\) matrix-valued function on \(\Omega \) with \(L^{\infty }\) coefficients. Consider the divergence-form operator \({{\mathscr {L}}}^{A}=-\mathrm{div}\,(A\nabla )\) with mixed boundary conditions on \(\Omega \). We extend the bilinear inequality that we proved in Carbonaro and Dragičević (J Eur Math Soc, to appear) in the special case when \(\Omega ={\mathbb {R}}^{d}\). As a consequence, we obtain that the solution to the parabolic problem \(u^{\prime }(t)+{{\mathscr {L}}}^{A}u(t)=f(t)\), \(u(0)=0\), has maximal regularity in \(L^{p}(\Omega )\), for all \(p>1\) such that A satisfies the p-ellipticity condition that we introduced in Carbonaro and Dragičević (to appear). This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on \(\Omega \), in particular, we do not assume any regularity of \(\partial \Omega \), nor the existence of a Sobolev embedding. The methods of Carbonaro and Dragičević (to appear) do not apply directly to the present case and a new argument is needed.



中文翻译:

不规则域上具有复系数的发散型算子的双线性嵌入

\(\ Omega \ subseteq {\ mathbb {R}} ^ {d} \)打开,并在\(\ Omega \)上使用\一个复杂的严格严格累加的\(d \ times d \)矩阵值函数设为A。(L ^ {\ infty} \)个系数。考虑散度形式的算子\({{\ mathscr {L}}} ^ {A} =-\ mathrm {div} \,(A \ nabla)\)\(\ Omega \)上具有混合边界条件。在特殊情况下,当\(\ Omega = {\ mathbb {R}} ^ {d} \)时,我们扩展了我们在Carbonaro和Dragičević(出现的J Eur Math Soc)中证明的双线性不等式。结果,我们得到抛物线问题\(u ^ {\ prime}(t)+ {{\ mathscr {L}}} ^ {A} u(t)= f(t)\)的解\(u(0)= 0 \),对于所有\(p> 1 \)\(L ^ {p}(\ Omega)\)中具有最大规则性,因此A满足我们引入的p-椭圆率条件在Carbonaro和Dragičević(出现)中。对于我们考虑的算子类别,此指数范围是最佳的。我们不对\(\ Omega \)施加任何条件,特别是,我们不假设\(\ partial \ Omega \)的任何规律性,也不存在Sobolev嵌入。Carbonaro和Dragičević(出现)的方法并不直接适用于本案,需要一个新的论点。

更新日期:2020-05-21
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