**2**

votes

**0**answers

178 views

### One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ...

**2**

votes

**0**answers

179 views

### Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...

**2**

votes

**0**answers

80 views

### What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?

Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set ...

**2**

votes

**0**answers

119 views

### Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$
Now, by ...

**2**

votes

**0**answers

84 views

### Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...

**2**

votes

**0**answers

39 views

### Hess-Schrader-Uhlenbrock inequality for non-symmetric operators

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some ...

**2**

votes

**0**answers

127 views

### Intuition behind Stokes operator?

I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is
$$A = -P_L Δ$$
where $Δ$ is the Laplacian, and $P_L$ is the Leray ...

**2**

votes

**0**answers

85 views

### Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)

Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...

**2**

votes

**0**answers

137 views

### Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...

**2**

votes

**0**answers

69 views

### How analyze the following fully nonlinear equation

Now I want to consider the following pde
$u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset ...

**2**

votes

**0**answers

99 views

### Uniqueness of solution of elliptic equation with exponential nonlinearity

Consider the following equation
$$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$
where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...

**2**

votes

**0**answers

73 views

### Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...

**2**

votes

**0**answers

99 views

### long time existence of a nonlinear parabolic equation

I am thinking about a geometric problem which boils down to the following parabolic equation:
Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$
$$\begin{cases}\displaystyle \frac{\partial ...

**2**

votes

**0**answers

223 views

### A integral equation with Discrete to result by inverse problem

Problem
I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...

**2**

votes

**0**answers

82 views

### Is logarithmic convexity of the heat kernel with complex time a general fact?

Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...

**2**

votes

**0**answers

165 views

### Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)

Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...

**2**

votes

**0**answers

131 views

### Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...

**2**

votes

**0**answers

96 views

### Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...

**2**

votes

**0**answers

124 views

### Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$.
Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...

**2**

votes

**0**answers

123 views

### Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems:
Let $f(x)$ be a convex funtional ...

**2**

votes

**0**answers

77 views

### Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation
$$
\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\
u(0)=u_{0}.
$$
with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...

**2**

votes

**0**answers

85 views

### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...

**2**

votes

**0**answers

126 views

### Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...

**2**

votes

**0**answers

72 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 ...

**2**

votes

**0**answers

131 views

### Newtonian potential for continuous $f$

Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$.
Consider the Newtonian potential of $f$ (at least I think this is what it is called)
$$ v(x)=\int_{R^N} ...

**2**

votes

**0**answers

62 views

### Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic).
The standard ...

**2**

votes

**0**answers

100 views

### Variant form of the gronwall inequality

I know the following statement for gronwall inequality:
Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have,
$f' \leq \phi f$ and $f(0)=0$ then $f=0$
Now is ...

**2**

votes

**0**answers

74 views

### Holder continuity of Poisson equation with divergence free drift

I am interested in the following PDE.
Suppose $u_m$ is a smooth solution of a elliptic equation of the form
$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on ...

**2**

votes

**0**answers

114 views

### Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...

**2**

votes

**0**answers

149 views

### Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$

Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$.
(Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$).
Let $v ...

**2**

votes

**0**answers

101 views

### Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$

Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...

**2**

votes

**0**answers

164 views

### Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...

**2**

votes

**0**answers

213 views

### $C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...

**2**

votes

**0**answers

107 views

### Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...

**2**

votes

**0**answers

82 views

### Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...

**2**

votes

**0**answers

185 views

### Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...

**2**

votes

**0**answers

137 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**2**

votes

**0**answers

109 views

### Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?

I would be interested to know if a certain set of periodic solutions for
the two-dimensional Navier-Stokes equations is closed generically.
Many similar (yet not identical) set-ups can be found in the ...

**2**

votes

**0**answers

143 views

### geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
...

**2**

votes

**0**answers

109 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 ...

**2**

votes

**0**answers

145 views

### Estimates on gradients of diffusion semigroups

Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form
$$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...

**2**

votes

**0**answers

98 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 ...

**2**

votes

**0**answers

77 views

### Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...

**2**

votes

**0**answers

98 views

### slightly subcritical elliptic pde; the linearized equations

Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems
$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...

**2**

votes

**0**answers

184 views

### Reference on a Monge-Ampère-like equation

We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$:
$$ \det(Hess(w))=1~, $$
...

**2**

votes

**0**answers

127 views

### Subordination identity and heat operator

I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...

**2**

votes

**0**answers

107 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 ...

**2**

votes

**0**answers

168 views

### 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) = ...

**2**

votes

**0**answers

105 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$, ...

**2**

votes

**0**answers

98 views

### 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 ...