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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functions of bounded variation and gradient vector measure
I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{...
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votes
1
answer
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$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...
- 1,051
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vote
0
answers
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Harmonic extension of $L^\infty$ function is in $L^\infty$?
Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = u(x)$$...
- 21
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votes
2
answers
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Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
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votes
2
answers
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Monge Ampere equations (concavity)
The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\...
- 3,383
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answer
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Maximum principle for heat equation on infinite domain
Let $u(x, t)$ be a solution of $u_t=u_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in ...
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1
answer
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Does this time-dependent trace space have a name?
This question is a follow up to this question.
Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in $H^{1/2}(\...
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0
answers
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Solution to a PDE with constant data - what is the fault in my proof? [closed]
Let $C=\Omega \times (0,\infty)$. We want to find a solution $v \in H^1(C)$ such that given $u \in H^{\frac 12}(\Omega)$,
$$\int_0^\infty\int_\Omega \nabla v \nabla \varphi + v_y\varphi_y = 0\quad\...
- 49
1
vote
1
answer
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Nonlinear parabolic PDEs existence with Galerkin method?
Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...
- 405
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vote
0
answers
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Negative Definiteness of Hopf-Lax-Oleinik Semigroup
Denote by $H_{t}$ the Hopf-Lax semigroup, i.e.\begin{equation}
H_{t}f(x)=\inf_{y\in\mathbb{R}}\left\lbrace f(y)+\frac{(x-y)^{2}}{2t}\right\rbrace.\end{equation} Is $H_{t}$ negative definite on bounded,...
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Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?
Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...
- 3,059
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1
answer
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Endpoint Strichartz Estimates for the Schrödinger Equation
The non-endpoint Strichartz estimates for the (linear) Schrödinger equation:
$$
\|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)}
$$
$$
2 \...
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Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional
In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:
\begin{equation}
\sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N\frac{\...
- 347
1
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1
answer
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A non-linear PDE
I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \...
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0
answers
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A variation of Poisson's equation in cylindrical coordinates
Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
...
2
votes
1
answer
192
views
Nonlocal Stefan problems
Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality?
A search on Google and MathSciNet give me only a handful of results which greatly ...
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0
answers
52
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From Boundary to righthandside
I have a problem coming from linear elasticity in $(x,y,z)\in\mathbb{R}^2\times \mathbb{R}^+$, $t\in \mathbb R$:
$$\left\{\begin{aligned}\partial_{tt} \sigma&=A(D_x,D_y,D_z) \sigma\\
\sigma\big|...
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3
votes
1
answer
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views
Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation
The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int_{\Omega} \nabla u \nabla v d x = \...
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votes
1
answer
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Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result
Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)
Let $\Omega \...
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Reference for Hodge decomposition
Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
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4
votes
1
answer
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reference for existence and blow up results in transport-like PDEs
This question was originally posted by me on math.stackexchange but I didn't get any answers and I thought that perhaps it would be better off here. I hope it's appropriate, I've encountered the ...
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3
votes
0
answers
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Strong solution to $u_t - \Delta_p u = f$
For $p > 1$, consider the equation
$$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$
$$u(0) = u_0$$
$$u|_{\partial\Omega} =0$$
for all $v \in W^{1,p}(...
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votes
0
answers
392
views
$C^0$ estimates in wrapped Lagrangian Floer cohomology
Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...
- 1,331
2
votes
0
answers
250
views
long time existence of a nonlinear parabolic equation
I am thinking about a geometric problem which boils down to the following parabolic equation:
Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$
$$\begin{cases}\displaystyle \frac{\partial u}{\...
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7
votes
1
answer
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views
Green functions on Riemann surfaces
Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...
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1
answer
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Solving Functional Equation
Continue with my previous question “Regarding Kolmogorov's Superposition Theorem”, here are some further questions:
Question-1
Is it true: for any given $C^1$ continuous real function $f(x, y): \Re^2 ...
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2
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1
answer
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Solvable PDE Problems [closed]
I'm currently in a PDE course where one of the requirements is to find a common PDE problem and explain how to solve it.
The problems found easily on google won't help me, since every student has to ...
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1
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1
answer
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views
Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert f\Vert_{...
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votes
2
answers
664
views
Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
0
votes
1
answer
123
views
The monotone operator in $BV$ space
I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
- 679
1
vote
0
answers
84
views
Hyperbolic PDE from total derivative?
Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...
3
votes
0
answers
96
views
Reconstructing a vector field on the circle
Consider a ODE on the circle of the form
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \omega(x) \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} \begin{pmatrix} x_1 \\ ...
user45183
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votes
1
answer
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views
Existence and estimates of a solution of a perturbed first order partial differential equation
My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you ...
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votes
1
answer
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Existence of solution with $u' \in L^2(0,T;L^2)$ to a nonlinear parabolic PDE
Consider the problem of finding $u \in L^2(0,T;H^1)$ with $u' \in L^2(0,T;L^2)$ such that
$$\int_0^T \int_{\Omega}u'(t)\varphi(t) + \int_0^T \int_{\Omega}\nabla (F(u(t)))\nabla \varphi(t) = \int_0^T \...
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Given a fixed convex domain $\Omega$ in 3D, for what value $c$ the function $f(c) := \int_{\partial \Omega} |x-c| d \sigma_x$ gets its minimum?
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^3$, then consider the following minimization problem:
$$\inf_{c \in \overline{\Omega}} f(c), \quad f(c) := \int_{\partial \Omega} |x-c| ...
- 1,350
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vote
0
answers
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views
Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
1
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1
answer
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sobolev embedding theorem in the smooth metric measure space
we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...
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votes
1
answer
149
views
When does $\{u\in H^1_0: \Delta_{\mu}u\in L^2\}=H_0^1\cap H^2$.
Let $(M,\mu)$ be a weighted Riemannian manifold. In Grigor’yan's book, he proves that the Dirichlet Laplacian $\Delta_{\mu}$ is self-adjoint on the set $u\in\{u\in H^1_0(M): \Delta_{\mu}u\in L^2\}$. ...
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0
answers
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views
Reference request: Weak harnack inequality for biharmonic equation
I have seen a lemma which I do not have any reference or hint for it.
Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and
let $u$ be a positive distributional supersolution to
...
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0
answers
128
views
A heat equation approach to the perturbation of vector field with center
Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...
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4
votes
1
answer
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views
Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism
The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p,...
- 5,824
5
votes
1
answer
418
views
Robin-Laplacian in unbounded domains
Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If $\Omega$...
0
votes
0
answers
47
views
$u$ satisfies Schrödinger equation implies $\mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \right) $ also?
Consider the Schrödinger equation (SE):
$i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0, (x, t)\in \mathbb R^{N}\times \mathbb R.$
$u(0,x)=\phi(x).$
Then, formally,
the solution of (SE) ...
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votes
0
answers
302
views
Log of heat kernel for positive time
A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and $p(t,x,...
- 705
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votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
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2
votes
0
answers
87
views
1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
3
votes
1
answer
340
views
A question on Schrodinger operator
I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be ...
2
votes
1
answer
212
views
The centralizer of Lienard equation
Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
- 366
3
votes
0
answers
127
views
Existence of at least one positive solution for semilinear biharmonic equation with critical exponent
Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{...
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votes
2
answers
644
views
Does there exists a necessary condition for Lp multiplier?
Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant $C$...
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