Improved Sobolev regularity for linear nonlocal equations with VMO coefficients
Simon Nowak
Abstract
Abstract This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In this paper, we follow this direction and show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions $$u \in W^{s,2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>W</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> in fact belong to $$W^{t,p}_{loc}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi>loc</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msubsup></mml:math> for any $$s \le t < \min \{2s,1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>≤</mml:mo><mml:mi>t</mml:mi><mml:mo><</mml:mo><mml:mo>min</mml:mo><mml:mo>{</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>}</mml:mo></mml:mrow></mml:math> , where $$p>2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> reflects the amount of integrability gained. In other words, our gain of differentiability does not depend on the amount of integrability we are able to gain. This extends numerous results in previous works, where either continuity of the coefficient was required or only an in general smaller gain of differentiability was proved.