Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
370
questions
5
votes
1
answer
338
views
Regularity up to the boundary for the Poisson problem
It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...
2
votes
1
answer
817
views
The Biharmonic Eigenvalue Problem on a Rectangle with Dirichlet Boundary Conditions
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\
&...
- 271
2
votes
1
answer
531
views
Does this PDE only have the trivial solution?
Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
- 366
7
votes
3
answers
1k
views
Does "solutions of an $n$-th order ODE form an $n$-dimensional vector space" somehow generalise to PDEs?
It is well known that the set of solutions $u:\mathbb{R}\rightarrow \mathbb{R}$ of an $n$-th order, linear, homogeneous ordinary differential equation
$$a_n(x)\frac{d^n u}{dx^n}+\dots + a_1(x)\frac{du}...
- 505
3
votes
1
answer
328
views
elliptic regularity of Neumann problem on Square
I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{...
- 1,363
2
votes
0
answers
139
views
elliptic regularity for Neumann BVP on square
I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(...
- 1,363
1
vote
0
answers
95
views
Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
- 5,217
2
votes
2
answers
692
views
Existence and uniqueness for two-dimensional time-dependent Schrödinger equation
I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions.
Unfortunately, I am a ...
- 181
8
votes
2
answers
398
views
Bounded input Bounded output stability for heat equation
This is a cross-post from Computational Science.
I am interested in proving or obtaining a counterexample to the following conjecture.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
- 142
8
votes
1
answer
483
views
Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?
Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
2
votes
0
answers
113
views
biharmonic equation with L^1 data and Navier Condition
I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
- 361
0
votes
1
answer
386
views
The hypoellipticity of a heat-like operator
I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac {...
- 5,217
12
votes
1
answer
470
views
Does hypoellipticity imply the existence of a parametrix?
Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is ...
- 843
0
votes
0
answers
108
views
solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,363
1
vote
1
answer
460
views
Elliptic pde with bilaplacian; boundary conditions.
I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ \...
- 1,363
0
votes
2
answers
175
views
Separation of variables for a particular PDE
Given the partial differential equation
\begin{equation}
(1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0
\...
- 333
10
votes
2
answers
377
views
Evolution operator for a linear parabolic equation
Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
- 528
5
votes
1
answer
1k
views
Gradient estimate for elliptic equation
Given:
1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$
2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$
3)$g=(g_i,\ldots,g_n)\in \mathcal{C}^\alpha(\...
- 61
1
vote
0
answers
402
views
Gilbarg-Trudinger's book Theorem 4.13
I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...
- 369
2
votes
1
answer
262
views
elliptic boundary regularity, tangential regularity
A have a question related to the boundary regularity of a solution of a Poisson equation on a bounded domain. But to make the question easier to pose I will state it on $ R_+^2:=\{ x \in R^2:x_2>0\...
- 1,363
4
votes
2
answers
354
views
Recognizing Schwartz regular distributions
Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...
- 5,217
2
votes
0
answers
67
views
Localized eigenfunctions of drift Laplacians
I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
- 21
10
votes
2
answers
1k
views
Well-posedness of Fokker-Planck equation
Consider the following equation on $[0,T]\times\mathbb{R}^n$
\begin{eqnarray}
&\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\
&\rho|_{t=0}=\rho^0,
\end{eqnarray}
where $V\in C^2(\...
- 339
6
votes
1
answer
587
views
How do solutions of a PDE depend on parameters?
Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^...
- 8,046
5
votes
3
answers
451
views
Structure of sign changes under the heat flow
Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
1
vote
1
answer
153
views
Nodal sets under the heat flow
Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of $...
1
vote
0
answers
48
views
Renormalization for Transport Equations with SBD velocity field
In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in SBD(...
- 11
1
vote
1
answer
204
views
Strong maximum principle for the heat equation in non-cylindrical domains
let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...
8
votes
1
answer
482
views
Mountain Pass theorem for minimization problems with constraints
Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see http://en....
4
votes
2
answers
438
views
Heat equation and evolution of number of critical points
Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
0
votes
0
answers
215
views
Solving gradient of an especial heat equation
In my research I came up with a gradient of heat equation on a edge-weighted graph as:
\begin{equation*}
\nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0
\end{equation*}
where ...
- 161
3
votes
1
answer
2k
views
Method of characteristics of a system of first order pdes
I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...
- 43
2
votes
0
answers
66
views
only solution to wave equation under certain restriction?
Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two bounded solutions to the wave equation
\begin{equation}
\partial_t^2u_i = \partial_x^2u_i
\end{equation}
obeying the restriction
$\...
- 1,451
2
votes
0
answers
432
views
Existence of Green's functions for PDEs
Here is what I think I know: Given a symmetric linear differential operator $\mathcal{L}$ that is positive definite on a function space(/space of distributions) $\mathcal{H}$, we can find its inverse, ...
- 141
1
vote
0
answers
170
views
Existence of solution?
I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let $\mathcal{...
- 131
2
votes
1
answer
594
views
Maximal minimum of Bessel functions
This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...
- 2,464
1
vote
1
answer
137
views
Wave equation with linear coefficients
The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...
- 153
2
votes
0
answers
141
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
- 43
1
vote
1
answer
163
views
Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$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_{...
- 13
1
vote
0
answers
94
views
References Request : Existence and Uniqueness for PDE which is "ALMOST (?)" Parabolic
I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case:
\begin{equation}
Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + \sum_{...
- 41
2
votes
2
answers
143
views
First order pde with characteristics [closed]
Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...
- 21
5
votes
1
answer
405
views
Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$
Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...
- 632
0
votes
1
answer
349
views
Holder regularity for the heat potentials
First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials.
If $\Gamma(t,x)$ is the fundamental ...
- 291
6
votes
1
answer
1k
views
Regularity of solution to Fokker Planck equation
Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t =...
- 339
1
vote
0
answers
136
views
A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11
2
votes
0
answers
110
views
Solve a PDE related to free boundary problem
I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
- 1,837
3
votes
1
answer
506
views
Uniqueness of weak solutions of a heat equation
Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...
- 33
3
votes
2
answers
261
views
Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1
In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
- 31
0
votes
0
answers
79
views
Finding gradient of an optimization
I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...
- 161
-2
votes
2
answers
208
views
Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]
Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
- 75