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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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      23 views

      Approximation of functions by tensor products

      Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
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      0answers
      43 views

      Approximation of functions in $L^p(R^d;L^\infty)$

      Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that $$...
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      votes
      1answer
      45 views

      Question on relation between a parabolic sobolev space and a sobolev bochner space

      For parabolic sobolev spaces I follow the following definition: According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$ ...
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      votes
      0answers
      62 views

      6 linear PDE for only 3 unknowns?

      Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
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      vote
      1answer
      92 views

      First order partial differential equation [on hold]

      I know there is a solution to this pde $$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$ $$ f(0,x)=g(x)$$ ( Where $v$ and $g$ are known functions) which is given by $$ f(t,x)=\frac{1}{v(x)} h(t+\...
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      Can there be a explicit expression of g as defined in the link

      This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...
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      votes
      0answers
      37 views

      Green’s function on upper unit half ball [on hold]

      Suppose we have this region in $\mathbb{R}^3$ $$ (x,y,z) \in \mathbb{R}^3, x^2+y^2+z^2<1, z>0$$ How to find Green’s function for this domain? And how it is possible to write the integral ...
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      31 views

      Neumann functions of Poisson problem [on hold]

      On page 219 of “Pinchover & Rubinstein” it is trying to find a function which is called Neumann function for $$ \Delta u= f, D$$ $$ \partial_n u= g, \partial D$$ It introduces $$h(x,y;\zeta,\eta)$...
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      votes
      1answer
      132 views

      Eigenfunctions of elliptic equations

      Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
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      0answers
      116 views
      +100

      Open Questions about Wasserstein Space and PDE

      While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
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      49 views

      What is the best book to learn about the wave equation? [closed]

      I'm looking for a book that teaches the wave equation and how to solve it for more advanced cases than the basic one (infinite/half infinite string, standing waves etc) What book would you recommend ...
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      vote
      1answer
      51 views

      Global solutions of the wave equation with bounded initial condition

      Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation $$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$ $$u(x,0)=f, ...
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      vote
      1answer
      79 views

      Two PDE for one unknown?

      Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My ...
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      votes
      2answers
      272 views

      Properties of heat equation

      ** I simplified the question: ** On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive. I ...
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      0answers
      70 views

      Measurability of the heat semigroup in $L^\infty$

      Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...

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