# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### How to find the associated conservation law from a given symmetry

It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/...
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### Poisson equation on exterior of a ball

Let $B_1^c$ denote the compliment of the unit ball centered at the origin in $R^N$ where $N \ge 3$. I am interested in $-\Delta u(x)=f(x)$ in $B_1^c$ with $u=0$ on $\partial B_1^c$. In ...
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### Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
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### Estimates on divergence-type operator for the matrix

Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form $${\rm div}(Av)=f$$ where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
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### A question about Stroock's notes on the Weyl lemma

On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
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The following is the PDE I want to solve, $$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}... 0answers 149 views ### Eigenvalue and eigenfunction convergence Consider a bounded Euclidean domain \Omega \subset \mathbb{R}^n (for simplicity, let's say, \Omega has smooth boundary and is simply connected). Let p \in \Omega be a point, and call \Omega_n = ... 1answer 105 views ### Interpolation for Sobolev spaces How one can identify the following (complex) interpolation space$$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$where \Omega is a regular domaine. After research, it seems that ... 0answers 56 views ### Proving that (f,g) are Cauchy data for the Schrödinger equation iff (f,g) satisfies an equation I have to prove that if f\in H^{1/2}(\partial\Omega) then (f,g) are Cauchy data for the Schrödinger equation if and only if$$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
I am currently looking at a $1$D-Burger's equation defined by \begin{equation} \label{ex burgers} \left\{ \begin{array}{ll} {} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...