We study three classical problems in the theory of linear second-order uniformly elliptic equations: (i) gradient estimates for the Dirichlet problem, (ii) estimates and asymptotics for the first eigenvalue of an elliptic operator, and (iii) the Landis conjecture on exponential decay. Our main results give versions of (i) in which the constant is sharply specified in terms of the norms of the coefficients of the operator and the size of the domain; and use these to strongly improve on known results for (ii) and (iii) by allowing both more general operators and weaker regularity assumptions on the coefficients, and by giving quantitative estimates. The proofs use a unified approach, relying on three ingredients: interior and boundary Harnack inequalities with sharp constants, duality arguments, and a $C^1$ estimate based on a rescaling procedure. The sharpness of the gradient and spectral estimates is demonstrated through various counterexamples.
We prove quantitative homogenization estimates for linear elliptic equations in nondivergence form with stationary, symmetric coefficients, finite range of dependence, and a deterministic upper ellipticity bound. No deterministic positive lower bound is imposed. We assume that the reciprocal of the infimum of the small...
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-co...
The presented work investigates the Cauchy problems for parabolic equations in both non-divergence and divergence forms with rough diffusion coefficients, which commonly arise in composite media, financial pricing, and viscous fluids. Under critical regularity settings, we establish the unique solvability in optimal fr...
We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is that, for compact classes preserved by the natural rescalings of the equation, qualitative and quantitative regularity have the same critic...
A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Mong...
Yifan Chen, Hans-Joachim Hein, Ryan Mc Gowan· 0 citations
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...
Wen-Guo Liang, Zheng-Ce Zhang· 0 citations
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