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4 votes

$W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations

The answer is yes if and only if you have optimal elliptic regularity in $W^{-1,p'}(B)$ for some $p>2$ for the adjoint differential operator, i.e., the one given by the transpose matrix $A^\top$. ...
  • 1,863
3 votes

An inequality for harmonic functions

Consider first the $d=2$ case. Then, $u$ is a real part of an analytic function. We can write $$u(z)=\frac12\sum_{n=0}^{\infty}(a_nz^n+\overline{a}_n\overline{z}^n)$$ and $$\partial_\nu u(z)=\frac12\...
  • 6,374
1 vote

A harmonic function degenerate in one direction

The questions have been answered in the comments, I am just recording them here: Alexandre Eremenko pointed out that no, the function $u$ need not be translation-invariant, because the dependencies on ...
  • 3,744
3 votes

'Dirichlet problem' along axis for harmonic functions

Assuming the Taylor series of $f$ has an infinite radius of convergence, the sum $$ \sum_{k=0}^\infty \left(x^2+y^2\right)^kf^{(2k)}(z)\cdot \frac{(-1)^k}{4^k k!^2} $$ converges absolutely and locally ...
  • 1,811

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