Regularity theory for non-autonomous problems with a priori assumptions
Peter Hästö, Jihoon Ok
Abstract
Abstract We study weak solutions and minimizers u of the non-autonomous problems $${\text {div}} A(x, Du)=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>div</mml:mtext> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$\min _v \int _\Omega F(x,Dv)\,dx$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>min</mml:mo> <mml:mi>v</mml:mi> </mml:msub> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>Ω</mml:mi> </mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> with quasi-isotropic ( p , q )-growth. We consider the case that u is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on A or F and the corresponding norm of u . We prove a Sobolev–Poincaré inequality, higher integrability and the Hölder continuity of u and Du . Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings. Connections between assumptions on A or F and assumptions on u are known for the double phase energy $$F(x, \xi )=|\xi |^p + a(x)|\xi |^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <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:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> </mml:math> . We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.