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This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an explicit estimate of the local $L^infty$-norm for the solutions gradient in terms of its local $L^p$-norm. Specifically, we prove begin{equation*} | abla u|_{L^infty(B_{frac{R}{2}}(x_0))}^p leq frac{C}{|B_R(x_0)|}int_{B_R(x_0)}| abla u(x)|^p dx. end{equation*} This estimate paves the way for our forthcoming work in establishing $W^{1,q}$-estimates (for $q>p$) for weak solutions to a much larger class of quasilinear elliptic equations.
For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for
This paper continues the development of regularity results for quasilinear measure data problems begin{align*} begin{cases} -mathrm{div}(A(x, abla u)) &= mu quad text{in} Omega, quad quad qquad u &=0 quad text{on} partial Omega, end{cases} end{a
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Ga
We prove in this paper the global Lorentz estimate in term of fractional-maximal function for gradient of weak solutions to a class of p-Laplace elliptic equations containing a non-negative Schrodinger potential which belongs to reverse Holder classe
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $displaystyle -operatorname{div}(A(| abla u|) abla u)+Bleft( | abla u|right) =f(u)$; in particular, we investigate the second order regularit