Parabolic and elliptic equations with singular or degenerate coefficients: The Dirichlet problem
Hongjie Dong, Tuoc Phan
Abstract
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Subscript plus Superscript d"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the coefficients are the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript d Superscript alpha Baseline comma alpha element-of left-parenthesis negative normal infinity comma 1 right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> <mml:mi> α </mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mi> α </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x_d^\alpha , \alpha \in (-\infty , 1),</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x Subscript d Baseline equals 0 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{x_d =0\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and they may not be locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.