MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider the linear second order elliptic Dirichlet problem

$$-\nabla\cdot (a\nabla u)\quad u=0 \text{ on }\partial\Omega$$

Condtion 1:$\Lambda |\xi|^2\geq\sum a_{i,j}(x) \xi_i \xi_j \geq \lambda |\xi|^2</math> for all <math>x\in \Omega, \xi \in \mathbb{R}^n$

The Schauder Estimates Boundary estimates yield

$|u|_{2,\alpha;\Omega} \leq C(n,\alpha,\lambda,\Lambda,\Omega) (|u|_{0,\Omega} + |f|_{0,\alpha;\Omega} + |\phi|_{2,\alpha;\partial\Omega}).$

Does this estimate hold uniformly for all $a$ that satisfy Condition 1 for a fixed $\lambda$ and $\Lambda$ and $||a||_{C^\alpha}\leq S$? If so what is the best reference to see this?

share|cite|improve this question

The Schauder estimates apply for the case when you consider the equation $$a_{ij}u_{ij} + b_i u_i + c u = f$$ and in this case you need to assume that there are bounds on the Holder norm of the $a_{ij}$. Your equation is of the divergence type, and there in this case your best result is a $C^{1,\alpha}$ bound, if $a_{ij}$ and $f$ are no better than $C^\alpha$. The simplest exposition of these concepts is probably to be found in the book of Qing Han and Fanghua Lin.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.