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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Alexandrov-Bakelmann-Pucci maximum principle
The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient ...
- 3,383
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The integrability of fundamental solution of laplace equation follows from integrability of f ?
Hi, I am really struggling with this question.
The question is :
Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta ...
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1
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Tangential boundary conditions for magnetostatic FEM problem
Hi everybody,
I am trying to solve a magnetostic problem with the Finite Element Method. But I have a problem applying tangential boundary conditions for the magentic field.
I solve for the vector ...
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Is this kernel space of finite dimension ?
Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
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Compactness of solutions of elliptic equation
Consider the following nonlinear elliptic equation
$$
-\triangle u + u + u^3 = g, \quad x \in R^3.
$$
If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
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1
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Basic questions about parabolic Holder space
Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...
CommunityBot
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2
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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 ...
CommunityBot
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Geometric Mean Value Property
Does anyone know where I could find a proof of a variant of a version of the mean-value property for harmonic functions in Riemannian manifolds? I'm actually more interested in using an elliptic ...
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3
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Classification of a certain System of Linear First Order PDEs whose characteristic polynomial has one real and two complex conjugate zeros
In my research work, I recently have come across the following system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros.
$3u_{1,1}+u_{3,1}...
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2
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Linear coupled parabolic PDE system with Holder continuous coefficients
I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$
$$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = g$...
- 16.2k
3
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ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
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how to solve a singular integral equation involving the kernel $1/x$
Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...
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Heat equation of spatial complex variable
Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation
$$\frac{\partial ...
CommunityBot
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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 ...
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3
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Question About Harmonic Function Theory
Given a non-negative function $u $ defined on $\mathbb{R}^2 $ , and satisfies :
$ \Delta u \leq 0 $ .
How can I prove that $u$ must be constant?
Is there an easy way to do it ?
Thanks !
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Integrability conditions for 'componentwise' systems of linear PDEs
I find myself staring blankly at a system of PDEs in $n$ dimensions which has "one equation per component" of the Hessian of the unknown function - that is, it specifies the Hessian in terms of the ...
- 108k
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1
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Wave equation v.s.Schrödinger equation
The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$
From the above that a wave operator can be ...
- 1,568
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1
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How to show this Holder bound?
Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \...
- 3,958
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Seeking reference on regularity theory for nonlinear elliptic PDE
Hello,
I am searching for a reference on a result I know must exist proving regularity for weak solutions of a (nonlinear, but well-behaved) elliptic homogeneous PDE. Working over say a bounded ...
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Recovering full regularity by energy method in the heat equation
Consider the heat equation
$$
u_t = u_{xx} + f,
$$
on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
- 3,322
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2
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Fundamental Solutions with compact support (distributions)
Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$
We also then argue that if a fundamental solution has compact support, then it is supported ...
- 1,644
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0
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215
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Coupled system of linear parabolic PDEs
Hi,
Are there any existence results for the coupled system of linear parabolic PDEs:
$$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$
$$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$
...
2
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1
answer
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Generalized Friedrichs Lemma
Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
- 16.2k
1
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1
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504
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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 ...
- 6,871
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1
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Reference: DaPrato and Grisvard parabolic PDEs.
Has anyone read G. DaPrato and P. Grisvard Equations d'evolution abstraites nonlineaires de type parabolique?
It's not available in my library. I am wondering if it's worth me acquiring it: is it ...
- 39k
2
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1
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687
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Solutions to Heat Equations with Obstacles!
Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \...
- 3,322
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Tensor analysis/Differential forms outside physics
There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...
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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-...
CommunityBot
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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 ...
CommunityBot
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Relation between interpolation spaces and besov spaces
Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x u\|_{L^{\...
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Existence of PDE system (mean curvature flow coupled with surface PDE)
Hi all,
What should I look for if I want to study existence/uniqueness of the system of PDEs:
$$u_t -\Delta u + u\nabla \cdot v = f(u) \quad\text{on $\Gamma(t)$}$$
$$X_t = \kappa N(X) + u \quad \text{...
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2
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The approximation to perturbed KdV Equation
Consider the perturbed KdV Equation $$u_t-6uu_x+u_{xxx}=\epsilon u$$,I want to use perturbative expansion to construct the solution as the form $$u=u(x,t;\epsilon)=\sum_{n=0}^\infty\epsilon^n u_n(x,t)$...
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0
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When can a perturbation be treated as a regular perturbation?
I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...
- 351
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0
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views
Convolution Estimates on a Smooth Manifold
Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying
$$
\|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty.
$$
Then ...
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1
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Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
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Proper sobolev spaces invariant under no-linearities
Let $f:H^s\to H^s$ at least continuous and not necesarily linear. Is there some kind of criterion or condition over $f$ that lets to ensure that $f({H^{s+k}})\subseteq H^{s+k}$?
CommunityBot
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Hölder estimates for the Complex Monge-Ampere equation
If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{...
- 6,871
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base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
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2
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Existence, uniqueness, and regularity for linear parabolic PDE on a complete Riemannian manifold
Let $M$ be a smooth manifold with a complete Riemannian metric $g$ and $E$ a smooth vector bundle over $M$ with an inner product and compatible connection $\nabla$. Let $K: E \rightarrow E$ be a ...
- 27.5k
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1
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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 ...
- 39k
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408
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Higher order partial derivatives and global regularity.
Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...
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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$ (...
- 6,891
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Pitfalls when generalizing the heat kernel of a Riemannian metric
Suppose $M$ is a Riemannian manifold with some compact quotient under isometries.
Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...
- 4,304
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0
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749
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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 ...
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3
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book on PDE on manifolds
let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...
- 39k
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1
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97
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Neumann problem in case f=1
Is there a solution to the following problem?
$-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$.
where $\Omega$ is bounded.
- 52.3k
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Asymptotic expansions of solutions of nonlinear PDE's
Let's consider the following Cauchy problem:
$$u_t=\alpha(x,t,u)u_{xx}+\beta(x,t,u)(u_x)^2+\gamma(x,t,u)u_x+\phi(x,t,u)$$
$$u(x,0)=u_0(x).$$
Assume that functions $\alpha,\beta,\gamma,\phi$ are ...
- 60.5k
1
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1
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regularity for fractional laplace equation
Hey, I want to know what is the best interior regularity of the following equaiton:
$(-\Delta)^{\frac{s}{2}}u=f$ in $B_{1}$ (ball with radius 1, centered at 0)
$f\in L^{\infty}(B_1)$
thanks
- 60.5k
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2
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742
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Hölder estimates on solutions of non-linear elliptic PDE.
In his book "Some non-linear problems in Riemannian geometry" T.
Aubin states the following result (Theorem 3.56):
Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order
differential ...
- 60.5k
2
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1
answer
557
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eigenfunction of heat operator.
Let $\Delta$ be the usual Laplacian on $\mathbb R^n$, $\Delta=-\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. Consider the heat operator $H_t=e^{-t\Delta}$. Is there an eigenfunction of $H_t$ which ...
- 52.3k