We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-coefficient harmonic replacement yields a sharp-function estimate for the Hessian, and the Fefferman--Stein and Hardy--Littlewood theorems permit the coefficient error to be absorbed. A parameter estimate, obtained by Agmon's auxiliary-variable argument, supplies a consistent resolvent on all $L^p$ spaces and makes the subsequent gain of integrability non-circular. The first-order term is retained throughout and is controlled by the scale-invariant quantity $R^{1-n/q}\|b\|_{L^q(B_R)}$, with $q>\max\{n,p\}$. As a consequence, coefficients in $\VMO_{\rm loc}$ give the usual local $W^{2,p}$ regularity for every finite $p$.
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality f...
In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for con...
We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,\gamma}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,\beta}$ estimates...
Aelson Sobral, Eduardo V. Teixeira, José Miguel Urbano· 0 citations
We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description...
Fernando Charro, M. del Mar González, María Soria-Carro· 0 citations
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