Compactness Methods for Certain Degenerate Elliptic Equations

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we obtain gradient continuity estimates for viscosity solutions of [Formula: see text] in terms of the scaling critical [Formula: see text]-norm of [Formula: see text], where [Formula: see text] is the normalized [Formula: see text]-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential [Formula: see text]. Moreover, for [Formula: see text] with [Formula: see text], we also obtain [Formula: see text] estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, [Formula: see text] regularity for the normalized [Formula: see text]-Poisson problem, J. Math. Pures Appl. (9) 108(4) (2017) 553–591], where a [Formula: see text] estimate was established depending on the [Formula: see text]-norm of [Formula: see text] under the additional restriction that [Formula: see text] and [Formula: see text]. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for [Formula: see text] continuous, our approach also gives a somewhat different proof of the [Formula: see text] regularity result.

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

OF THE DISSERTATION On some nonlocal elliptic and parabolic equations by Tianling Jin Dissertation Director: YanYan Li We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem involving nonlocal conformally invariant operators. Regularity properties for solutions of some degenerate elliptic equations as well as a Liouville type theorem are established, and used in our blow up analysis. We also introduce a fractional Yamabe flow and show that on the conformal spheres (Sn, [gSn ]) it converges to the standard sphere up to a Mobius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations, which are further used to improve a Sobolev inequality via a quantitative estimate of the remainder term.

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Congested traffic dynamics, weak flows and very degenerate elliptic equations