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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Estimating a potential
Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but ...
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Better regularity than $C^{1,\alpha}$ for $p$-harmonic maps into certain target manifolds
This is a follow up question I had while reading through this question and "Representing Homotopy Groups and Spaces of Maps by $p$-harmonic maps" by Shihshu Walter Wei.
We know that ...
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Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
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Atiyah-Singer fake proof intuition?
Here is one formulation of the Atiyah theorem.
Fix a Riemannian $M$ and a pde $P: E \to F$ of bundles.
Let $\pi: S^*M \to M$ be the cotangent sphere.
Let $\sigma$ be the index of $P$ (so swap ...
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Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying
$$
\begin{...
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Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?
If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
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Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)
What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
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On an estimate in the paper by Donnelly and Fefferman
I was reading the following paper by Donnelly and Fefferman https://link.springer.com/content/pdf/10.1007/BF01393691.pdf which essentially deals with the Hausdorff dimension bound of the nodal sets ...
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Image of trace operator on $W^{2,1}(\mathbb{R}^2)$
It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$.
For ...
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On the weak derivative of $|u|^{(p-2)/2}u$
Let $u$ be a function such that $|u|^{(p-2)/2}u$ is in $H^1_0(G)$, $G$ is open and $p>2$.
How can I show that
$$
D(|u|^{(p-2)/2}u)=p/2|u|^{(p-2)/2}D(u) \label{1}\tag{1}
$$
or how can I show that, ...
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Derive elliptic maximum principle from weak derivatives
Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that
$$...
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Can we deduce $a_n\le Cn^2$ from $a_{n+1}\le Ca_n+Ca_n^{1/2}$? [closed]
Assume $\{a_n\}$ is a positive sequence satisfying $a_1\le C$ and $a_{n+1}\le C a_n+C a_n^{1/2}$. Can we deduce by induction $a_n\le Cn^2$? Here C's could be different constants.
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Eigenvalue and eigenfunction convergence
Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
CommunityBot
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Well posedness of the Plateau problem under lack of uniqueness
The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not.
Premises
I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
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References for well-posedness of weak solutions to Stefan problem
Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
...
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Question concerning some calculation in the paper by Figalli and Neumayer
I am trying to read the paper by Figalli and Neumayer with the DOI 10.4171/JEMS/837 [arXiv:1510.02119]. I am having some trouble understanding the spectral analysis part. It would be very helpful if ...
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A question about the eigenfunction method and the notion of solution - distributional solution
I have a question about how a passage was made in the calculation of passage (2.5) in the calculation below. To introduce context, the author in the paper (full work) is trying to demonstrate that ...
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Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
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A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
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Question about Neumann eigenvalues on manifolds
Question:
Let $\Omega\subset \mathbb{S}^2_+$ be any geodesically convex subset of the hemisphere $\mathbb{S}^2_+$. Then is it true that $\mu_1(\Omega)\geq \mu_1(\mathbb{S}^2_+)$ where $\mu_1$ is the ...
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Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
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Why are viscosity solutions useful solutions?
I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.
...
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Feynman–Kac formula for other operators
I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$
where $x \in \Omega$ and $t > 0$, then
$e^{t\Delta_D}f(x) = ...
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A follow up on "A bilinear estimate with a simple one-dimensional oscillatory integral kernel"
I am investigating the validity of the estimate
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,dz\leq C\,\epsilon\, \|f\|...
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Regularity of solution to Fokker Planck equation
Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t =...
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Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?
I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...
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Strong convergence of differential quotient in $L^2(0,T;V^*)$
I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
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Techniques to estimate PDE which are elliptic in some directions and degenerate in others
I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
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How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
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Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
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Classification of homogeneous distributions
On page 92 of these notes, there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good ...
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Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
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Trouble understanding Lax method for KDV equation for inverse scattering method
I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
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Hörmander's hypoellipticity theorem for complex coefficients
Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...
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Complex sum of squares of vector fields (hypoelliptic operators)
Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$
Now, by ...
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Some improper integral
While studying problems related to fractional Laplacians $(-\Delta)^s$, I keep come across with type of quantities such as
$$
\int_0^R r^{2k}f(r)\,dr+\int_R^\infty r^{2k}\varphi(r)\,dr-\sum_{j=0}^\...
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
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PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
CommunityBot
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Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
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An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
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When considering an equation on flat torus, can we treat it as an equation on a square with opposite sites topologically identified? [migrated]
I need to use some theorems (which is originally for $2$-dim bounded domain in Euclidean space) on torus, what should I do to the equation? Can I just simply treat it as an equation on a square with ...
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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem
This is the equation given ($n\geq2$)
$$
\begin{cases}
u_{tt}=a^{2}\left(\Delta u\right), \\
\left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\
\left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) .
\end{...
CommunityBot
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An expansion for 2d Euler equation
Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$:
$$
-\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
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How to understand the unique continuation result
Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm
$$
\|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}.
$$
Suppose $K(x) \in C^1\left(\mathbf{R}^...
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Continuity of the constant in maximal Sobolev regularity
Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
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Specific type of PDE
While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices):
$$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
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Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?
Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$:
\begin{equation}
u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1}
\end{equation}
Assume that $...
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Chain rule for distributional derivative
Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$).
Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
CommunityBot
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$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.
Consider the problem
$$\Delta u = f \quad\text{in $\...
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