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.
4,467 questions
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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 $\...
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How does electric potential relate to mean curvature?
Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.
Now consider the electric potential generated by this uniform mass ...
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Existence of Green's function and the Dirichlet problem
Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \...
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Focusing NLS: $L^2$ convergence of a solution as $t\rightarrow +\infty$
Consider the cubic focusing non linear Schrodinger equation in dimension $n\geq 2$:
$$(iu_t+\Delta)=-|u|^2u\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}^n)$$
Can we find an initial data $u_0\neq 0$ such ...
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Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]
In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
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Can a function be constructed from the direction of its gradient?
Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...
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localization of the $L^p$ variation for heat equation
I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is.
The ...
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Pseudoinverse of Neumann-Laplacian
Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ~\mathrm{d}\...
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Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness
Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...
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Sobolev Inequality
Let $\Omega$ be a bounded region in $R^n$ and define
$W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$
where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that
$||u||_{L^2(\Omega)...
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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:...
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How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets
$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
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How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?
We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$.
More ...
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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 ...
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Distributional limits concerning the regularity of Maxwells equations
This question is related to my previous question about the regularity of the Maxwell equations.
Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...
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is this a known method for solving PDEs
I recently posed a system of PDEs to solve on MSE at https://math.stackexchange.com/q/514147/36530. It was quickly solved by a nice pair of subsitutions.
However, in this post, I'd like to show here ...
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Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
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Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$
Let $D\subset R^d$ be a bounded Lipschitz domain. We know that the Neumann eigenfunction lies in $C(\overline{D})$ (i.e. continuous up to the boundary). This can be seen from the fact that $\phi_k=e^{...
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A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]
Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) \...
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Finding an $H^1$ function given its values on $\partial\Omega$
Background
I've met this problem when I was trying to convert a elliptic PDE problem
into the corresponding variational problem in order to apply finite element method.
The PDE is an elliptic PDE ...
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Null sets in PDE
Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$,
$$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \...
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Hypoellipticity of square root of laplacian
It is a well known result (sometimes called the Weyl lemma) that the laplacian in $\mathbb{R}^n$ is hypoelliptic, i.e. if $f$ is a distribution s.t. $\triangle(f)$ is smooth in an open set, than $f$ ...
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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 ...
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Finding singular "solutions" to the Dirichlet problem for Schrödinger operators that do not admit smooth solutions
Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.
Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon ...
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When to use more exciting function spaces than ordinary Sobolev spaces?
In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces $L^2(0,T;H^...
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vector valued pde's good reference
I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows.
Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n \...
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parabolic PDE with pseudomonotone operators
I am looking for a reference where well-posedness of problems
$$u_t + A(t)u = f$$
is addressed via the Galerkin method where $A$ is a pseudomonotone operator. I am aware that Roubicek's book ...
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If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
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Non symmetric coefficient matrix for elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form
$$ D_i(a_{i,j}D_ju)=0 \tag{1}$$
where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
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PDEs as a tool in other domains in mathematics
According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...
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Nonlinear PDE and Green functions
This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
\partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real ...
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Applications of PDE in mathematical subjects other than geometry & topology
Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau ...
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Must Neuman Elliptic operator has discrete spectrum ?
It is well known that the Neuman eigenvalue problem has discrete spectrum and the eigen values are
nonnegative and can be arranged in a nondecreasing order of magnitude.
Do we need any smoothness ...
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(sharp)Garding's inequality and inequality with lower bounds
The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part $$P_{2m}...
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W^2,p regularity for solutions of elliptic equations
I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...
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Fully non-linear PDE
A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
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radial limits of subharmonic functions
Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every ...
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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 ...
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Does a particular iteration produce a weak solution to a non linear pde?
Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...
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A contradiction to do with continuity? (involves chain rule)
Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE
$$\frac{d}{dt}D^0_t(\cdot) = V(D^...
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The extension of smooth function
If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
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Blow up of solutions to parabolic PDEs
I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice ...
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Applications of pseudodifferential operators to PDE
I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole ...
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What are the advantage of using operational calculus for numerically solving pde compared to FE or FD?
For numerically solving a partial differential equation (PDE) what advantage does operational calculus (OC) has over common methods like finite difference (FD), and finite element (FE)?
I mean OC in ...
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Changing the space of test functions in PDEs
Let $V \subset H \subset V'$ be a Hilbert triple.
We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\...
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Boundary flux maximizing drift (velocity) vector fields for 2D heat equation
Looking for literature / known results on the following class of problems:
Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...
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Lebesgue Riemann Theorem.
Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that
Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x)...
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$L^2$ boundeness of a sequence
Let $f_n \in C^2(\bar{\Omega})$ be a sequence satisfying
$\Delta f_n - f_n^3 \to 0 \ \ {\rm in} \ \ L^2(\Omega)$
where $\Omega \subset {\mathbb R}^2$ is bounded and open with a smooth boundary. Is ...
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A Sobolev-type inequality with weights
In the study of a particular PDE I found myself wanting to prove the following inequality:
$( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2]...
- 60.5k
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Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
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