Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
2answers
215 views

Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
2
votes
1answer
130 views

Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction. Imagine one has two evolution equations $$\partial_t u = p(x,\partial_x,f)u$$ $$\...
2
votes
1answer
130 views

Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
2
votes
0answers
57 views

Smooth dependence of convex functions on Monge-Ampère measure

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as ...
0
votes
2answers
80 views

Reference request for fractional Poincare inequality

Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
3
votes
1answer
70 views

Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let $f : \...
0
votes
1answer
86 views

Separation of variables for PDE

Consider the PDE $$ \partial_t f(t,x) = \Phi(x) f(t,x)+ \Psi(x)f(t,x-a)$$ $$f(0,x)=1$$ $$f(t,0)=1$$ where $a$ is a constant and $\Phi$ and $\Psi$ are some differentiable functions. To solve this I ...
2
votes
1answer
223 views

Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
3
votes
1answer
116 views

Liouville theorem for fractional Laplacian

Is there any Liouville type theorem for the half space problem \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\ v & =0 &&\text{...
0
votes
0answers
46 views

linear fractional laplacian problem

If $U(x)$ is a classical solution of \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s U+ U &=m(r)U &&\text{in } B \\ U &= 0 &&\text{in } \mathbb R^N \...
1
vote
1answer
75 views

Extended Global approximation theorem

In Evans, $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
3
votes
1answer
215 views

Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
4
votes
1answer
134 views

Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
1
vote
1answer
99 views

Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
3
votes
1answer
48 views

Difference between linear and parabolic velocity profiles in Stokes flows of two fluids

For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a). ...
4
votes
1answer
103 views

Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...
1
vote
1answer
178 views

Boundary regularity for the Monge-Ampère equation $\det D^2u=1$

Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying $$ \det D^2u=1,\quad u|_{\partial\Delta}=0. $$ Classical ...
1
vote
0answers
60 views

Unique continuation from the boundary for inhomogeneous elliptic pde

Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
3
votes
1answer
180 views

Stationary Navier-Stokes solutions

Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
1
vote
0answers
72 views

asymptotic behaviour of principal eigenfunctions and Large Deviations

Dear Math Overflowers, I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
1
vote
0answers
48 views

On the non-existence of weak solutions to nonlinear evolutionary PDEs without time derivative

Imagine we have a nonlinear PDE, e.g. some velocity model $$\begin{aligned} -\Delta v + (v \cdot \nabla) v + \nabla p = f \\ \text{div} \, v = 0 \end{aligned}$$ which has a weak solution for some ...
1
vote
1answer
101 views

Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put ...
1
vote
1answer
114 views

Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results. Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
9
votes
2answers
439 views

Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
14
votes
3answers
327 views

What is known about sufficient conditions for the rigidity of a convex surface?

A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$. An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
4
votes
2answers
388 views

Klein Gordon equation - references

The Klein Gordon equation of the form: $\Delta u+ \lambda u^p=0$ is been studied for $p = 2$? (i.e.$\Delta u+ \lambda u^2=0$) If yes are there references?
2
votes
0answers
75 views

geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...
0
votes
0answers
40 views

What are tools we need to prove scattering result for NLS when propagator is unitary?

Consider nonlinear Schrödinger equation (NLS) $$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$ where $F$ is some nonlinearity. Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...
1
vote
0answers
53 views

Global interior estimate complex Monge-Ampere equation

Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
3
votes
1answer
339 views

Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
0
votes
1answer
140 views

Regularity in Orlicz spaces for the Poisson equation

I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007) $$\Delta u=f \quad \quad \quad \quad (1)$$ Lemma 2: There is a constant $N_1 >1$ so that ...
4
votes
1answer
378 views

Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
1
vote
0answers
57 views

Elliptic pde L^p theory via adjoint theory

Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...
1
vote
0answers
86 views

Weak elliptic maximum principle on manifolds without strict ellipticity

This question is not to be confused with the similarly titled question here. In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...
0
votes
1answer
91 views

Is $X = \{ B \in L^\infty(\mathbb R^n,\mathbb R^n): \nabla \cdot B \in L^\infty(\mathbb R^n,\mathbb R^n) \}$ a dense subspace?

The Sobolev space $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ is not dense in $L^\infty(\mathbb R^n,\mathbb R^n)$. In fact the functions in $W^{1,\infty}(\mathbb R^n,\mathbb R^n)$ are Lipshitz, and not ...
2
votes
0answers
70 views

Wave equation with 'spring' integral boundary condition

I am really stuck with this small toy problem. I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is: ...
0
votes
0answers
38 views

Interpolation inequalities involving mean curvature operator

Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator $$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$ (in any ...
-5
votes
1answer
275 views

Big list of “outstanding paper awards” [closed]

What are the prizes awarded for significant papers published in mathematical journals? One of them is the SIAM Outstanding Paper Prize. I'd be particularly interested in hearing about prizes ...
2
votes
0answers
59 views

Reference on elliptic obstacle problem that covers the material in the lecture notes by Caffarelli

Can you recommend a modern, self-contained, readable reference that covers (approximately) the results on elliptic obstacle problem that are covered in the lecture notes by L. Caffarelli (Scuola ...
1
vote
0answers
102 views

Solution of nonlinear heat equation decreases in time

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a smooth decreasing function which is non-negative and bounded above, and consider the heat equation on a smooth bounded domain $$u_t - \Delta u = f(u)$$ ...
11
votes
2answers
496 views

Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post. Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there ...
2
votes
0answers
89 views

Ricci flow with surgery without the “no locally separating $\Bbb RP^2$” assumption

In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ ...
3
votes
0answers
83 views

Ratio of solutions to two heat equations

Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
1
vote
2answers
58 views

Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...
0
votes
0answers
77 views

any hope of $C^{2,\alpha}$ regularity for linear problem

Let $B_1$ denote the unit ball in $ R^N$ (say $N \ge 3$) and consider $$ \Delta \phi + \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} \phi_{x_i x_j} = f $$ in $B_1$ with $ \phi=0$ on $ \partial B_1$. ...
7
votes
2answers
607 views

Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
0
votes
0answers
25 views

regularity of function from spherical representation

I asked a question regarding the linear operator a few days ago here is this explicit linear operator hypo-elliptic . The operator in question was $ L(\phi)=\Delta \phi(x) + \gamma \phi_{rr}$. We ...
1
vote
0answers
59 views

Using semigroup theory for nonautonomous semilinear equations

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
1
vote
0answers
44 views

Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
3
votes
2answers
247 views

Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Where can I find a (readable and self-contained) proof of the following result? Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...