Gradient estimate for solutions of second-order elliptic equations
نویسندگان
چکیده
We obtain a local estimate for the gradient of solutions to second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at single point $x=0$. In particular, we treat case not Lipschitz show our is sharp.
منابع مشابه
Boundary Estimates for Positive Solutions to Second Order Elliptic Equations
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions.
متن کاملPositive solutions to second order semi-linear elliptic equations
Here G ⊆ R (N ≥ 2) is an unbounded domain, and L is a second-order elliptic operator. We mainly confine ourselves to the cases F (x, u) = W (x)u with real p and W (x) a real valued function on G, and F (x, u) = g(u) with g : R→ R continuous and g(0) = 0. The operator L = H − V is of Schrödinger type, namely V = V (x) is a real potential and H = −∆ or more generally H = −∇ · a · ∇ is a second or...
متن کاملSecond-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
The aim of this work is to revisit viscosity solutions’ theory for second-order elliptic integrodifferential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new Jensen-Ishii’s Lemma for integro-differential equations, which is stated for solutions with no restriction on their growth at infinity. The pr...
متن کاملMaximum Norm Estimate for Bivariate Spline Solutions to Second Order Elliptic Partial Differential Equations in Non-divergence Form
The convergence of the bivariate spline solution to the solution of the second order elliptic PDE in non-divergence form in the maximum norm is presented in this paper. Mainly, the L∞ norm of the spline projection in the Sobolev space H 0 (Ω) ∩H 0 (Ω) is shown to be bounded, where Ω is a polygonal domain. With the boundedness of the projection, one can establish the error of the spline solution...
متن کاملLinear Elliptic Equations of Second Order
The classical Schauder type results on C-regularity of solutions are exposed for linear second order elliptic equations with Hölder coefficients. Our approach is based on equivalent seminorms in Hölder spaces C, which are similar to seminorms introduced by S. Campanato [1]. Under this approach, the C-estimates for solutions are derived from the maximum principle and the interior smoothness of h...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Advances in Differential Equations
سال: 2022
ISSN: ['1079-9389']
DOI: https://doi.org/10.57262/ade027-0102-77