# 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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### Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field" Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$ Consider the ...
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### Dirichlet problem for manifold, how to prove $W^{1,2}_0(\Omega)$ solution is $C^{2,\alpha}(\bar{\Omega})$?

Let $M$ be an n dimensional Riemannian manifold without boundary. Let $\Omega \subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(\bar{\Omega})$, Consider the Dirichlet problem. ...
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### Neumann equation on manifold with edge or corner

Let $(M,g)$ be a compact Riemannian manifold with boudnary and corner, i.e. locally mdoelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$. ...
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### What is the geometric or dynamic meaning of a global attractor with an infinite fractal dimension?

In Efendiev-Ôtani's article: Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium (Ann.I. H. Poincaré, AN 28,2011), is obtained that the fractal dimension ...
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### Partial regularity of harmonic maps into spheres

Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
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### A condition for Laplacian

Let $u\in L^{2}(\mathbb{R}^{2})$ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$. Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$? Thank you in advance.
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### Is there a connection between representation theory and PDEs?

As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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### Does the Leibniz (product) rule hold for the spectral fractional Laplacian?

Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
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### Regularity of solution to a hyperbolic pde

I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^2(0,T;V^*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a ...
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### Interpolation Inequality's Proof

Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$: \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
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### PDE system problem to find the metric

I have a big problem to solve this system: $\Delta f-hf^2=0$ $p|\nabla f|^2+hf^3=0$ where $h$ and $p$ are constants (with $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on ...
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### Spectrum of Dirichlet Problem for Laplacian on a Parallelogram

Let $M \subset \mathbb{R}^2$ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and 120)....
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### A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
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### Heat equation on torus

Consider heat equation on torus: $$\partial_tu(x,t) + (- \Delta)^{\alpha/2} u(x,t)=0, u (x, 0)=u_0(x)$$ where $(x, t) \in \mathbb T^d \times \mathbb R, \alpha>0$ Formally, we may write the ...
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### Traces of manifold-valued Sobolev maps

Let $(M^m,g)$ be a compact Riemannian manifold with smooth nonempty boundary, and $N^n\subseteq \Bbb R^d$ a boundaryless isometrically embedded Riemannian manifold. For $1\le p<\infty$ we define as ...
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### Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold: $$\lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$ I do not have access to his 1967 two papers,...
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### A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let \$\Delta_{\...