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In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small bal …

$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\| …

Let $ \Omega $ be a bounded domain with smooth boundary. Consider the Poisson equation \begin{eqnarray} -\Delta u&=&f\text{ in }\Omega\\ u&=&0\text{ on }\partial\Omega \end{eqnarray} where $ f\in C_0^ …

In the book "Regularity Theory for elliptic PDE", written by Fernández-Real, page 67, $ \tilde{u}_{k} $ converge to $ \tilde{u} $ only in $ C^1 $ norm, but the result is that we can get a equation for …

For the elliptic equation with non-divergence form $$ \sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1), $$ where $ \{a_{ij}(x)\} $ is a matr …

I am reading some papers on Green functions of elliptic equations. Here the elliptic systems is stated as $ Lu=-\operatorname{div}(A\nabla u) $ where $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix val …

Recently I am reading a book of elliptic equations. In the beginning there is a famous Caccioppli inequality for weak solutions. The theorem is stated as follows Suppose that $ u\in H^1(B(0,1)) $ sat …

Let $ \Omega\subset\mathbb{R}^d $ be a Lipschitz domain and $ A=(a_{ij}(y)):\Omega\to\mathbb{R}^{d\times d} $ be a matrix valued function with uniformly elliptic conditions i.e. $ \lambda|\xi|^2\leq a …

Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega, …

Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\alpha, …

I have read a paper of Z. Shen [1]. In the paper the author mentioned we can deal with two-dimensional elliptic systems by adding a dummy variable (the method of ascending) and use the results on the …

Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\alpha, …

Dirichlet problem for Laplace equation as follows \begin{eqnarray} \Delta{u}&=&0\text{ in }B_r(0)\\ u&=&g\text{ on }\partial B_{r}(0), \end{eqnarray} where $ g $ is continuous. It is already known t …

I come across an interesting question. Let $ B_r=\{x\in\mathbb{R}^3:|x|\leq r\} $ be the ball in $ \mathbb{R}^3 $ with radius $ r $. Assume that $ u \in C(\mathbb{R}^3\setminus B_1) $ satisfies $$ \D …

Recently, I am learning Schauder estimates for elliptic equations and I come across a proposition as follows Let $ \alpha\in (0,1) $ and $ \Omega $ be a bounded $ C^2(\Omega) $ domain on $ \mathbb{R} …