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
6
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Harmonic maps into compact Lie groups
Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck,
Schoen, Richard; Uhlenbeck, ...
0
votes
1
answer
531
views
Question on PDE
I am looking for a reference where the following situation or something similar could have been studied. As a foreword, my question may not be very technical since I am from an engineering background. ...
- 13.9k
3
votes
0
answers
179
views
How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded
With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider
$$u' + Au = f$$
in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
- 41
1
vote
0
answers
58
views
sharp conditions characterizing the vanishing of scalar Jacobi fields
Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$.
I was ...
- 156
2
votes
0
answers
263
views
Best constant of Gagliardo-Nirenberg inequality in exterier domain
In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have
$||u||_4^4\leq C||u||_2||\nabla u||_2^...
- 43
5
votes
1
answer
298
views
A nonlinear system with special structure
Suppose we have an $n \times n$ uniform grid, covering $[-1,1] \times [-1,1]$ (typically, n $\approx$ 500). We have a smooth, differentiable function $z(x,y)$ that we want to determine on the nodes of ...
- 51
1
vote
0
answers
99
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Limit Toward Discontinuous Point of Dirichlet Boundary Value
The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...
- 291
4
votes
1
answer
634
views
Higher order Sobolev inequality
Let $(M,g)$ be a closed, Riemannian manifold of dimension $n>4$. Let $K$ be the best constant for the Sobolev inequality
$||u||^2_p \leq K \int_{{\Bbb{R}}^n} (\Delta u)^2 dx,$
where $p=\frac{2n}{...
- 1,701
2
votes
2
answers
569
views
smoothness of solution for second order elliptic problem
Hello all,
could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem
$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$
$u = g \;,\; x \...
- 53
1
vote
0
answers
153
views
How to pick out harmonics based on boundary conditions?
(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...
- 1,893
3
votes
0
answers
42
views
Does the fast diffusion equation (or singular PME) on a manifold lose mass if the exponent is small enough?
Consider the singular porous medium equation
$$u_t - \Delta (|u|^{m-1}u) = 0$$
$$u(0)=u_0$$
given $u_0$ bounded, where $m \in (0,1)$.
When posed on $\mathbb{R}^n$, it is well known that mass is ...
- 91
2
votes
0
answers
128
views
How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?
I am reading this paper.
Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$
On page 5 of ...
- 23
3
votes
1
answer
225
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meromorphic family of pseudo-differential operators
Let $(M,g)$ be a closed, Riemannian manifold. Let $S(z)$ be a holomorphic family of pseudo-differential operators, with $z \in \Bbb{C}$. Let $u$ be a smooth function. Does it follow that $\lim_{y \...
- 1,701
1
vote
1
answer
630
views
Stuck on a convergence argument in $H_0^1(\Omega)$.
I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.
However I've encountered this step along the way which seems clear to me ...
- 2,641
1
vote
2
answers
421
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Showing a solution of elliptic PDe is non-degenerate
Dear Mathoverflowers:
I am interested in radial positive solutions of
$-\Delta u(r) = r^\alpha u(r)^p$ in the unit ball in $ R^N$ with $ u=0$ on the boundary.
Here $p>1$ and $ \alpha >0$. (...
- 13
1
vote
1
answer
259
views
$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?
I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
...
- 5,327
3
votes
1
answer
452
views
Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.
A friend of mine in the department needs to know if the following PDE has been extensively studied
$$ u_t = (u^2)_{xx}$$
Or more generally, replacing the square by any function of $u$. One would like ...
- 4,466
1
vote
0
answers
71
views
Bogomol’nyi’s Formula for the Critical Action
I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is
$(|F_A|^2+|d_A\phi|^2+|\frac12(1-|\phi|^2)|^2)...
- 31
1
vote
1
answer
231
views
Is there such a priori estimates for mean curvature type equation?
I am dealing with a mean curvature type equation as following:
$\displaystyle{\sum_{i,j=1}^{2}}(\delta_{ij}-\frac{u_{i}u_{j}}{1+|Du|^{2}})u_{ij}=(1+|Du|^{2})^{\frac{1}{2}-\frac{1}{2\alpha}}$, where $\...
- 215
0
votes
0
answers
303
views
hitting probability for integrated Ornstein-Uhlenbeck process
Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...
1
vote
0
answers
525
views
How to prove that 1 is not an eigenvalue of $T'(x)$?
Given a compact continuous operator $T$ from a Banach space $V_1$ to itself and $T$ maps a convex closed bounded set $\mathcal{B}$ into itself, how can we show that 1 is not an eigenvalue of $T'(x)$ (...
3
votes
1
answer
471
views
Regarding Discrete Eigenvalues
For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete.
But, ...
- 3,383
4
votes
0
answers
309
views
Recovering full regularity by energy method in the heat equation
Consider the heat equation
$$
u_t = u_{xx} + f,
$$
on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
- 3,322
2
votes
1
answer
621
views
What happens to the boundary conditions as a PDE is approximated by a lesser order PDE?
Consider a fourth order linear (biharmonic) PDE in two variables of the form
$\nabla^4u + c\nabla^2u-\lambda u = F(x,y)$; $(x,y) \in \Lambda$
To have uniqueness, we must specify two equations per ...
- 543
1
vote
0
answers
37
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Regularity of a flux induced by a potential
Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a scalar ...
- 71
2
votes
0
answers
420
views
Variational Formulation of Boundary Value Problems With Unknown on the boundary
Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,
\begin{eqnarray}
Lu &=& \frac{\partial u}{\partial t},
u(x,0) &=& u_0 \; \; \...
user20631
2
votes
0
answers
170
views
Elliptic equations with divergence-free drift terms
Given
$\
\mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ bounded, $div$$(\mathbf{u})=0$, $\...
- 171
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answers
153
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Critical elliptic equation; kernel of linearized operator
I am interested in the critical equation
$- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin (...
- 539
2
votes
0
answers
159
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Helmhotz decomposition and Regularity in Stokes equation
It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely
decomposed as
\begin{eqnarray*}
\
f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)}
\
\end{eqnarray*} with $f_{0}\in L_{...
- 171
3
votes
0
answers
128
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$L^2$ bounds for the gradient of subsolutions to parabolic equation
Suppose we have the differential inequality
$$
|\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|)
$$
in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...
- 879
2
votes
0
answers
151
views
Why pseudoconvexity is important in Partial differential equation theory?
I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
- 308
1
vote
0
answers
103
views
Regularity of weak solutions for a quasilinear problem
Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in C^{1,\beta}_0(\...
- 11
0
votes
0
answers
214
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Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
- 101
2
votes
2
answers
1k
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Application of coordinate-stretching transformation for Perfectly Matched Layer
A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems.
PML note
The PML ...
- 157
0
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0
answers
60
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The minimizing problem over a sequence of shrinking balls
Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define
$$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$
where $1<q<5$. Hence we know that each $\...
- 679
2
votes
0
answers
157
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linear operator associated with semilinear elliptic pde
I am reading a paper where at some point they analyse the following linear operator:
$$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$
where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
- 539
2
votes
2
answers
517
views
Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity?
It is known that the solution of equation
$$
u_t - \triangle u = \kappa|u|^{\sigma}u, u(0) = u_0
$$
blow up in finite times if $\sigma > 0$. That is, the $L^{\infty}$ norm of solution $u$ will goes ...
- 81
3
votes
1
answer
427
views
Spectral Galerkin method for a semi-linear parabolic PDE
I'm trying to understand how to apply the Galerkin method to $u_t - \Delta u = u^3$. I understand how to obtain all of the a-priori estimates using Sobolev embeddings and such but my question concerns ...
- 2,641
0
votes
1
answer
309
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Nonlinear PDE ${u_{tt}}^2u_{ttxx} = 1$
I have been trying to solve this equation during fortnight
$$
{u_{tt}}^2u_{ttxx} = 1.
$$
But I still here. The only thing is change of variables $u_{tt}(t,x) = y(t,x) $ and solved the ODE $y'' = \frac{...
- 399
2
votes
0
answers
103
views
Almost a Green formula
Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
- 171
2
votes
1
answer
492
views
PDEs on the Klein bottle and real projective plane
I would appreciate a reference describing the analysis of PDEs on the Klein bottle and the real projective plane. As an example, is there a reference discussing the existence and uniqueness of the ...
- 103
1
vote
0
answers
142
views
Elliptic problem on half space; infinite boundary values; Liouville theorem
In a the study of a boundary value problem the following problem is arising:
$-\Delta v(x)= e^{v(x)}$ in $ R^N_+$
$v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$.
...
- 539
2
votes
1
answer
186
views
ellipticity independent of metric?
I am new to the theory of pseudo-differential operators on compact manifolds, but I need to use a result related to this theory in a proof I'm working on. The problem is as follows: Let $(M,g)$ be a ...
- 1,701
7
votes
0
answers
438
views
Regularity of solutions to a linear degenerate parabolic pde
I've encountered the following problem which is causing me some trouble :
Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function $u:...
- 4,123
1
vote
1
answer
234
views
Continuation of hyperbolic Laplacian eigenfunction
The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen 79/3-...
- 3,337
1
vote
1
answer
468
views
A more accurate method to solve hyperbolic PDE
Hi all,
I have a set of hyperbolic PDE and I have been solving this equation uisng Lax-Wendroff method (from Richtmyer). The solution is OK while I am looking for a better approach to do it. Is there ...
- 31
1
vote
0
answers
205
views
Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
1
vote
1
answer
186
views
Well-posedness of Euler-Poisson system for semiconductors
Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? ...
- 35
6
votes
1
answer
578
views
Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)
The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
- 8,597
0
votes
0
answers
88
views
References for LWP of a NLS Equation
I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...
- 516