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.
4,466 questions
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N-soliton, The Lax operator and the transmission coefficient
I'm interested in the soliton stability result given in HERBERT KOCH and DANIEL TATARU's paper
"MULTISOLITONS FOR THE CUBIC NLS IN 1-D AND THEIR
STABILITY", published in IHES. However, I ...
- 705
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answer
308
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A certain solution for Sine-Gordon Equation
I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
- 245
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Nodal domain theorem for clamped plate equation
Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the clamped plate equation in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...
- 87
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2
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On a 3D Gagliardo-Nirenberg inequality
It is well known that there exists a constant $C$ such that
$$\forall f\in C^\infty_c(\mathbb R^3), \quad
\Vert f\Vert_{L^6(\mathbb R^3)}\le C\Vert \nabla f\Vert_{L^2(\mathbb R^3)}.
\tag{$\ast$}$$
Now ...
- 16.2k
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Comparison principle for a type of PDE with Neumann condition on a unbounded domain
We define a domain
$$\Gamma = \{(x_1, x_2, y) \in \mathbf R^3: x_1 < y\}.$$
Consider a PDE with $(x_1, x_2, y) := (x, y)$
$$\frac{1}{2} \Delta_x v(x, y) + b(x) \cdot D_x v(x, y) - v(x, y) + L(x) = ...
- 1,399
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Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
- 457
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$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form
$$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$
are there any elliptic ...
- 93
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$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line
I consider the following problem on the half real line
$$
\begin{cases}
u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm]
u|...
- 43
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Finiteness of the moments of the Malliavin derivative of the stochastic heat equation
I am studying section 2.4.2 from Nualart's book "The Malliavin calculus and related topics" on the stochastic heat equation. I have some questions on the validity of some estimates for the ...
- 61
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55
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Compactness and Leray-Schauder degree
What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
- 471
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78
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Nonlinear quadratic Schrödinger equation with variable coefficients
Consider the following quadratic Schrödinger equation in $Q_T = \Omega \times [0,T]$:
$$\begin{cases}
i \partial_t u + \kappa(x,t) \Delta u + \beta(x,t) u^2 = f(x,t)\\
u(x,0) = ...
- 111
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Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form
$$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$
...
3
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Extension of Sobolev function defined on unit cube
Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
- 93
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Some questions on a paper of Rellich
I was trying to read the paper "Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u =0$ in unendlichen Gebieten" by Franz Rellich (MR17816, Zbl 0028.16401).
Since it is in ...
- 41
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Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
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219
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Compactly supported wave packet in Schrödinger's evolution
Does the following result hold:
For any compactly supported wave packet, under free Schrödinger's evolution, it is no longer compactly supported after any finite time?
6
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1
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206
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Reference for $\epsilon$-regularity
I am looking for a reference for the following $\epsilon$-regularity statement: let
$(M,g)$ be a Riemannian manifold of dimension $n$,
$\Delta=dd^*+d^*d$,
$B_r$ denotes a ball of radius $r$ around a ...
- 954
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Are all solutions to the linear heat equation $\partial_t u - \Delta u = 0, u(0,\cdot) = 0$ continuous at $t = 0$?
Consider a distributional solution $u(t,\cdot) \in C^0([0,T],\mathcal{D}'(\mathbb R^n))$ to the linear heat equation
$$
\left\{
\begin{align*}
u_t - \Delta u &= 0, \\
u(0,\cdot) &= 0
\end{...
- 233
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1
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Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions
I was looking at Theorem 12.4 of Gilbarg and Trudinger's Elliptic partial differential equations of second order (MR1814364, Zbl 1042.35002):
Theorem 12.4. Let $u$ be a bounded $C^2(\Omega)$ solution ...
- 537
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184
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Bounded functions satisfying $\Delta u \geq u$ on $\mathbb{R}^n$
Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that
$$
\begin{cases}
u(x) > 0 & \forall x \in \mathbb{R}^n \\
\Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n
\end{...
3
votes
1
answer
187
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Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
8
votes
0
answers
115
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optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
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1
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90
views
PDE where the square of gradient of the unknown equals a given positive function
Let $V(x)$ be a non-negative smooth function defined in a open domain $U\subset\mathbb{R}^n$. Suppose that $V(x)=0$ only at a given point $x_0\in U$. Consider the PDE
$$|\nabla u|^2=V$$
with ...
- 13
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3
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313
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Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 1,759
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113
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Stability of perturbation of an elliptic problem into a parabolic PDE
Fix some $f\in H^1(\partial (0,1)^d)$.
Let $\eta\ge 0$ and for each such $\eta$ consider the solution $u^{\eta}$ to the solution PDE
$$
\begin{cases}
\Delta u & = u_t - \eta u_{tt} \mbox{ on } \...
- 5,405
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votes
2
answers
364
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Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
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2
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Growth of nonnegative functions satisfying $\Delta u \geq C>0$
Let $u\in C^2(\Omega)$ with $\Omega = B_1(0)\subseteq \mathbb{R}^d$. Assume that $u$ is subharmonic and satisfies the inequality
$$ \Delta u(x) \geq C>0 $$
for all $x\in \Omega$. Furthermore, we ...
- 1,045
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1
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580
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Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
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249
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The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by
$$Lu = \partial_i(a^{ij}...
- 91
3
votes
1
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377
views
A more general product rule for weak derivatives?
Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
- 1,759
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A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
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0
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114
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SPDE Renormalization
some SPDE (in higher dimensions) can only be interpreted in a "renormalised" sense. For example considering $\Phi_2^4$ on $\mathbb{R}_+\times \mathbb{T}^d$ the solution is defined as the ...
- 573
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votes
1
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How many non-trivial solutions can a semilinear elliptic equation have on a smooth star-shaped bounded domain with 0-Dirichlet boundary conditions?
I am not an expert in elliptic partial differential equations, but while studying the attractor structure of evolutionary PDEs, I frequently encounter problems related to elliptic equations. ...
1
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2
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237
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Calderón–Zygmund/$L^p$ estimates for the linear heat equation
Let $C_r$ denote the open cylinder
$$
C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\}
$$
and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation
$$
\...
- 233
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votes
1
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There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?
Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
- 677
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584
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Reference request: Software for producing sounds of drums of specified shapes
Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
11
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3
answers
727
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Application of Lie group analysis of PDE (beyond calculation of exact solutions)
I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide ...
- 111
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0
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90
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Euler-Lagrange equation of fractional Laplacian
The following result is in "An extension problem related to the fractional Laplacian" Section 3.2 by Caffarelli-Silvestre. I’m confused how to show it and wish to have some help.
Suppose $u:\...
- 421
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93
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$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE
$$\partial_t u=\Delta u-u^3+\xi,$$
where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
- 1,904
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1
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75
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How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$
Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$.
Assume ...
- 101
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Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
- 1,429
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1
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297
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Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 2,818
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
2
votes
0
answers
90
views
Positivity for a kinetic PDE
Let us consider the following kinetic equation:
$$ \partial_t f + v \cdot \partial_x f = \rho[f] \, M[T] - f $$
for a the phase space density $f=f(x,v,t)$ on $\mathbb{T}^1 \times \mathbb{R} \times (0, ...
- 185
2
votes
2
answers
154
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Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
- 1,759
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,313
3
votes
1
answer
176
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Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
- 1,759
1
vote
0
answers
52
views
High order parabolic PDEs on manifolds: Reference request
I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
- 363
6
votes
1
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228
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Question about Bochner measurability
When I study parabolic pde's I often came across the following type of Bochner spaces $L^p([a,b];L^{q}(\Omega),\ W^{1,p}([a,b];L^{q}(\Omega))$ and $L^{q}([a,b];W^{1,p}(\Omega))$ where $p,q\geq 1$ and $...
- 1,759
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votes
0
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3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$.
Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
