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Results tagged with real-analysis
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user 68370
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
4
votes
0
answers
296
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And supp …
- 261
3
votes
1
answer
141
views
PDE satisfied by projection of a function onto a subspace
Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$ …
- 261
2
votes
1
answer
907
views
One side Harnack inequality for Subharmonic function
It is well known that for any non negative Harmonic function w ($\Delta w=0$, $w\geq 0$) in a ball, $B_1(0)$, $\exists$, C>0 such that $\forall y\in B_{1/2}(0)$
$$
Cw(0)\leq w (y)
$$
It is a clear i …
- 261
0
votes
1
answer
106
views
Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_...
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1 …
- 261