Articles of elliptic equations

Is there a Maximum Principle for Biharmonic eigenvalue problem?

Let $$\Delta^2 u-\lambda u =0$$ where $\lambda>0$ and $$\Delta^2 u = \frac{\partial ^4 u }{\partial x^4} + 2 \frac{\partial^4 u }{\partial x^2 \partial y^2} + \frac{\partial ^4 u }{\partial y^4}$$ Is there a maximum principle for this equation in a rectangular domain $[-a,a]\times[-b,b]$ or any other domain? Fact: it is known that there is a […]

Can one obtain Neumann boundary conditions via Friedrichs extension?

According to the book “Applied Functional Analysis” vol. I by Zeidler, the Friedrichs extension of an operator $B\colon D(B)\subset X\to X$, where $X$ is a real Hilbert space, is obtained as follows. One needs: that $B$ be symmetric, that is $(Bu, v)=(u, Bv)$ for all $u, v\in D(B)$; that $B$ be strongly monotone (aka strictly […]

Why is it useful to show the existence and uniqueness of solution for a PDE?

Don’t get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps for example, to actually solve the problem. I am reading books that deal with variational approach for elliptic PDEs like the Laplacian. Apart from transforming the […]

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset \mathbb{R}^n$ be an open set of class $C^2$ with $\partial \Omega$ bounded. Let $f \in L^2(\Omega)$ and $u \in H_0^1(\Omega)$ satisfies the weak formulation […]

Unique weak solution to the biharmonic equation

I am attempting to solve some problems from Evans, I need some help with the following question. Suppose $u\in H^2_0(\Omega)$, where $\Omega$ is open, bounded subset of $\mathbb{R}^n$. How can I solve the biharmonic equation $$\begin{cases} \Delta^2u=f \quad\text{in } \Omega, \\ u =\frac {\partial u } {\partial n }=0\quad \text{on }\partial\Omega. \end{cases} $$ where $n$ […]