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        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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        A particular semi-linear equation on Riemannian manifolds

        Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
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        34 views

        Strong maximum principle for fractional laplacian

        Consider the problem $$(-\Delta)^{s} u+ u\geq 0 \text{ in } \Omega $$ and $u\geq 0 \text{ in } \mathbb R^N \setminus\Omega.$ If $u$ is continuous upto the boundary of $\Omega$, is it true that $u>0$...
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        41 views

        PDE on an open ball with prescribed value on some open subsets

        Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
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        49 views

        Existence and smoothness for viscous Burgers equation?

        What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?
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        27 views

        A Piecewise PDE

        Any idea for solving a piecewise PDE in form $\partial_{t} f(t,x)$ =\begin{cases} \partial_{x}[(ax+b)f(t,x)] & x\leq \alpha \\ \partial_{x}[(cx^2+dx+e)f(t,x)] & \alpha\leq x\leq \...
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        142 views

        Laplacian spectrum asymptotics in neck stretching

        Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
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        Symmetry of solution of fractional operator

        Is it true that any classical positive solution of $$(-\Delta)^su=f(u) \text{ when } x\in (-1, 1), u=0 \text{ when } |x|\geq 1 $$ is even. I am aware of the result of symmetry in higher dimension.
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        118 views

        Divergence form degenerate pde and Feynman Kac

        Consider $$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$ and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
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        A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

        Let $M$ be a Riemannian manifold with boundary $\partial M$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
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        105 views

        Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

        $$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$ Other than integrate this term by term (which might look crazy)? Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...
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        1answer
        119 views

        Practical applications of Sobolev spaces

        What are the examples of practical applications of Sobolev spaces? The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of ...
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        72 views

        Localization of solutions for time-dependent Schroedinger equation

        I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know. The ...
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        57 views

        Estimate of $L^1$-norm of semilinear elliptic inequality

        Let $\Omega$ be $\mathbb R^n$ or a complete non-compact manifold, we consider $$\Delta u+f\cdot u+u^2\leq0,$$ where $\Delta$ denotes $-\sum^n_{i=1}\partial^2_{x_i}$ and $f$ is a $C^2$ function such ...
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        Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

        Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...
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        37 views

        A first order PDE

        Does anyone know if there is a solution for this PDE : $$ \partial_t P(t,x)= \partial_x [(a+b e^{-x}) P(t,x)]$$ $$ P(0,x)=f(x)$$ Where $a$ and $b$ are constant and $f(x)$ is a known function?

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