Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation
Yuzhou Fang, Vicenţiu D. Rădulescu, Chao Zhang
Abstract
Abstract We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$\begin{aligned} -{\text {div}}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty ,\ a(x)\ge 0. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mtext>div</mml:mtext> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> We find some appropriate hypotheses on the coefficient a ( x ), the exponents p , q and the nonlinear term f to show that the viscosity solutions with a priori Lipschitz continuity are weak solutions of such equation by virtue of the $$\inf $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>inf</mml:mo> </mml:math> ( $$\sup $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>sup</mml:mo> </mml:math> )-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest.