# 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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### Can Schauder's fixed point theorem apply to a metric space?

I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
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### Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...
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### Proof of uniqueness of solution of the Poisson's equation for given boundary conditions [closed]

I asked this question last week on Mathematics Stack Exchange, but I didn't get any feedback. It is not a research level question, however, I post it here, perhaps somebody could help me: I would ...
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### propagation of a invariance along some PDE

Consider the following non linear PDE on $\mathbb{R}^n$ $$\partial_t u_t(x) \,=\, F\big(x, u_t(x), D u_t(x)\big)$$ with given initial condition $u_0(x)$. Assume that: $u_0$ is rotation invariant, ...
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### Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain $$\begin{cases} \rho_t + q_x = 0 \\ q_t + (q^2/\rho + \rho)_x = - q \\ u|_{t=0} = u_0 \\ u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t) \end{cases}$$ in which ...
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### A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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### Reference on noncommutative PDE

I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
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### Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
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### An inequality for the Hessian of eigenfunctions of the Laplacian on compact manifolds

Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator. Let $\lambda_1 >0$ be the first positive eigenvalue. That is, there exists a non-trivial function ...
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### Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator

I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why ...
Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?