# 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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### Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
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### Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces

Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
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### Weighted Sobolev norm in terms of Spherical harmonics coefficients

Let $M = [1,\infty) \times S^2$. Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm: $$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV$$ ...
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### Is the single layer potential associated to the operator $-\mathrm{div}(A \nabla)$ a $H^{-1/2}$-elliptic operator?

Let $A$ be a $3\times 3$ symmetric definite positive real constant matrix, $\Omega\subset \mathbb{R}^3$ a bounded lipschitz domain, $\Gamma$ its boundary and $g$ the fundamental solution of the second ...
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### Is Poisson formula valid for the weak solution of Laplacian?

In the book "Regularity Theory for Elliptic PDE", here is a theorem as follows Theorem(Harnack's inequality). Assume $u\in H^1(B_1)$ is a non-negative, is the weak solution for the ...
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### How to prove the regular result of elliptic equation $-\operatorname{div}(A(x)\triangledown u)=F$ in $B(0,1)$ if $A$ has the Holder regularity?

Consider the elliptic equation $-\operatorname{div}(A(x)\triangledown u)=F$ in $B(0,1)$ the ball with center $0$ and radius $1$ in $\mathbb{R}^d$, where $F\in L^p(B(0,1);\mathbb{R})$ for ...
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### $L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs: $$u_t - \Delta u = f$$ $$\partial_\nu u = 0$$ $$u|_{t=0} = u_0$$ where $f \in L^p(0,T;L^r(\Omega))$ and \$...