This research examines the impact of fractional Laplacian on weak L^p-solutions in elliptic equations, suggesting crucial implications for regularity.
We consider an elliptic equation with the fractional Laplacian operator (-Δ)α/2 in the dissipative term, a singular integral operator A(·) in the nonlinear term, and an external source f. The key example is the stationary (time-independent) counterpart of the surface quasi-geostrophic equation. Under suitable assumptions on f and natural assumptions on A(·) in the setting of Sobolev spaces, our main result examines how the fractional power α propagates and optimally improves the regularity of weak Lᵖ-solutions to this equation.
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Oscar Jarrín (2025) studied this question.
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