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
4
votes
1
answer
283
views
Lower bound on the solution of a Schrödinger-type equation
Consider the equation
$-\Delta u + Vu=f$,
on a closed manifold (or on a bounded domain with homogeneous Neumann condition). Here one can assume whatever integrability or smoothness conditions on $V$ ...
- 3,322
1
vote
0
answers
88
views
Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$
Consider the equation
$$u' + Au = f$$
$$u|_{\partial \Omega} = 0$$
$$u(0) = u_0$$
where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
- 93
4
votes
0
answers
343
views
Tangential boundary regularity for optimal transport maps
I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II:
Assume $u$ is a convex ...
- 4,899
4
votes
0
answers
217
views
Manifolds with a lower degree of regularity
I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below).
There, the authors consider domains of $R^n$ with regularity of class $W^2 L^{n-1,1}$(...
- 351
2
votes
0
answers
237
views
Reference on a Monge-Ampère-like equation
We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$:
$$ \det(Hess(w))=1~, $$
...
- 1,730
1
vote
1
answer
198
views
Solution formular for Laplace equation [closed]
I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads:
$-\Delta u = f$ in $R^3_+$,
$\partial_3 u|_{x_3=0}=g$ on $R^2$.
If $f$, $g$ satisfy compatibility ...
- 19
1
vote
1
answer
511
views
Heat equation of spatial complex variable
Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation
$$\frac{\partial ...
- 19
0
votes
1
answer
348
views
Dual space of Bochner space: is there an easier proof to show they're isometric?
It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$.
If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for the general case ...
2
votes
0
answers
149
views
Variational inequality on Manifold
Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla \...
- 21
1
vote
0
answers
108
views
A bilinear estimate in Lp space
Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that
\begin{equation}
\|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*)
\end{...
- 425
1
vote
1
answer
1k
views
A priori energy estimates for Burger's equation with dissipation
I've prove existence using the Galerkin method forBurger's equation with dissipation:
$u_t + uu_x - u_{xx} = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.
Clarification: I have ...
- 2,641
2
votes
1
answer
546
views
Surface PDE (heat equation) weak form and existence/uniqueness
Suppose $X:A \subset \mathbb{R}^n \to \mathbb{R}^{n+1}$ is a parametrisation of a surface $\Gamma$ such that $X(\theta_1, \theta_2) = (\theta_1, \theta_2, h(\theta_1, \theta_2))$ (i.e., $\Gamma$ is a ...
- 45
2
votes
1
answer
449
views
Wavefront set of a product
Let $H$ be the Heaviside function. If $f(x_1,x_2)=H(x_2)$ on $\mathbb{R}^2$, then $WF(f)=N^*\{x_2=0\}$. Similarly, if $g(x_1,x_2)=H(x_1^2-x_2)$, I think the wavefront set of $g$ is the conormal ...
- 35
1
vote
0
answers
198
views
Passing to the limit in a PDE (subsequence problems)
For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$...
- 155
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
0
votes
1
answer
100
views
properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$
Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for $|\...
- 1,051
4
votes
0
answers
257
views
What's a good resource for Hormander symbols of type (1/2, 1/2)?
I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...
- 545
3
votes
0
answers
146
views
Variational Principle for a System of Differential Equations
I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v \...
- 516
3
votes
2
answers
642
views
Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
0
votes
0
answers
1k
views
characteristic surface
Hello,
I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ :
(1) $G_{xy}=0$
(2) $G_{xz}=0$
(3) $G_{yz}=0$
(4) $G_{xx}-G_{yy}=0$.
It is not hard to see that the general ...
1
vote
0
answers
91
views
Existence for special Dirichlet problem
I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...
- 11
3
votes
1
answer
258
views
Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 1,211
3
votes
1
answer
309
views
ellipticity and invertible differential operators
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}_+$. Suppose that the differential equation $Pu=f$ has a unique $H^r(M)...
- 1,701
0
votes
1
answer
181
views
What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?
Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $...
- 1,471
2
votes
0
answers
749
views
Pullback of harmonic forms.
If $f \colon X \to Y$ is a holomorphic map between Kaehler manifolds, then the pullback of a harmonic form on $Y$ is not necessarily harmonic on $X$, even if $f$ is an immersion. This came up during a ...
- 101
3
votes
0
answers
381
views
Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
4
votes
1
answer
261
views
Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
user7807
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
3
votes
1
answer
653
views
Elliptic estimates and regularity of the $\overline{\partial}$-operator with totally real boundary conditions in $W^{1,p},$ $1<p\le 2$
Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:
(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-...
- 2,935
2
votes
0
answers
202
views
geometric irregularities in pde's
The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
http://link.springer.com/...
- 113
4
votes
1
answer
159
views
diffusions corresponding to estimators
I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
- 43
4
votes
1
answer
297
views
a question about Lp norm of curvature on convex curves
Suppose we have two strictly convex closed curves $C_{1}$ and $C_{2}$, $C_{1}$ contains $C_{2}$,
then can we conclude $\int_{C_{1}} \kappa_{1}^{p} ds\geq \int_{C_{2}} \kappa_{2}^{p} ds$, $\kappa_{1}$ ...
- 215
3
votes
1
answer
302
views
Stability of Dirichlet data for Helmholtz equation
I'm dealing with the Helmholtz equation $\Delta u +k^2u=0$ in a exterior region $R^3/D$ ( $D$ opened and bounded) of a three dimensional space with Dirichlet boundary condition $u=g$ on $\partial D$ (...
- 35
1
vote
0
answers
267
views
Extension of solutions of PDE
Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let $F:...
- 11
3
votes
0
answers
119
views
Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?
Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of
$$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$
where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...
- 107
2
votes
0
answers
155
views
Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)
Denote by $\mathbb{E}(g)$ the solution of the PDE
$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$
$$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$
Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...
- 155
1
vote
1
answer
2k
views
FFT setting the boundary conditions
I am using the FFT method to solve the two dimensional PDE on a $2\pi\times 2\pi$ square. I have boundary conditions that $u(x,0,t)=0$ and $u(x,2\pi,t)=0$. How do I go about setting these boundaries? ...
- 3,217
3
votes
0
answers
693
views
Convolution Estimates on a Smooth Manifold
Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying
$$
\|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty.
$$
Then ...
- 31
1
vote
1
answer
504
views
W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation
In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
http://arxiv.org/abs/1111.7207
They proved the $W^{2,1}$ estimate for standard monge-ampere equation
$detD^{2}u=f$
with $f$ bounded from below ...
- 13
4
votes
0
answers
453
views
Existence of solutions to elliptic PDE in undbounded domain
Specifically, I have $Lu=f$, where $L$ is a linear elliptic pseudodifferential operator, on an unbounded domain of the form $\Omega\times [0,\infty)$, $\Omega$ has Lipschitz boundary, $u$ is 0 on all ...
- 41
3
votes
0
answers
178
views
Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?
I would be interested to know if a certain set of periodic solutions for
the two-dimensional Navier-Stokes equations is closed generically.
Many similar (yet not identical) set-ups can be found in the ...
1
vote
1
answer
259
views
Does this PDE has a general solution? [closed]
$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$
K and u are constants.
If no,how to get a asymptotic solution?ie,linearize
- 689
1
vote
0
answers
232
views
Invariance of a tensor Laplacian
Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let $...
- 3,322
4
votes
1
answer
280
views
Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$
Assume that P is a real valued strong elliptic polynomial, then what do we know about the following
$$
K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0
$$
The ...
- 1,644
2
votes
2
answers
153
views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
- 101
3
votes
1
answer
138
views
Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?
Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow
$$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = 1?$$...
- 363
3
votes
1
answer
603
views
Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$
Hi,
i know that the following statement is used extensively, but i cannot find a proof anywhere:
For $\Omega$ a Lipschitz domain with boundary $\Gamma$, the space $H^{1/2}(\Gamma)$ is dense in $L_2(\...
- 33
0
votes
1
answer
275
views
On the fractional Schrödinger equation
I wonder if there is any theory about what we can call the fractional Schrödinger equation:
$$
\mathrm{i}\frac{\partial \psi}{\partial t} = (-\Delta)^s \psi + g(|\psi|^2)\psi \quad\hbox{in $\mathbb{R}^...
- 459
0
votes
0
answers
103
views
How to prove it is uniformly bounded?
Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
- 41
2
votes
0
answers
146
views
Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$
Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...
- 5,380