**1**

vote

**2**answers

95 views

### Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...

**1**

vote

**0**answers

78 views

### Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising:
$-\Delta v(x)= e^{v(x)}$ in $ R^N_+$
$v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$.
...

**0**

votes

**1**answer

108 views

### Application of Toms- Stein restriction theorem for Strichartz estimates

The initial value problem for one dimensional Shrödinger equation is
$$iu_{t}+u_{xx}=0,$$
$$u(x, 0)= f(x),$$
where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued ...

**3**

votes

**0**answers

188 views

### Uniqueness theorems related to Hardy Uncertainty Principle

Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that
if $f(x)=O(e^{-\alpha^2|x|^2})$, ...

**0**

votes

**1**answer

124 views

### Proof of global (in time) existence of classical solutions for 2D Euler equation in bounded domain

Anyone can explain the main idea, or recommend some paper or book on that?
For the whole space case, or the periodic case, the proofs are everywhere. But those do not seem to apply to the bounded ...

**2**

votes

**1**answer

222 views

### unique continuation property for overdetermined elliptic PDE

On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is ...

**1**

vote

**1**answer

205 views

### Reference request: Spectral analysis of advection diffusion PDE

As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the ...

**0**

votes

**1**answer

103 views

### analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem
Lu = 0 in R^n,
where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...

**3**

votes

**0**answers

96 views

### Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...

**3**

votes

**1**answer

227 views

### A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two ...

**2**

votes

**0**answers

81 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$, ...

**4**

votes

**2**answers

254 views

### Reference Request: Probability and (Nonlinear) PDEs

I'm a graduate student interested in learning about probability and (mostly evolutionary) PDEs, just for fun (and as an excuse to learn some probability). I'm mostly interested in things along the ...

**1**

vote

**2**answers

306 views

### Reference request: learn measure theory for PDEs

I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...

**1**

vote

**2**answers

134 views

### Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial ...

**2**

votes

**2**answers

112 views

### 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 ...

**1**

vote

**0**answers

174 views

### local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE:
$u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$
with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...

**1**

vote

**0**answers

62 views

### Kernel of perturbation of biharmonic operator

Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form:
$$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$
with ...

**3**

votes

**0**answers

93 views

### Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation
$- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin ...

**1**

vote

**1**answer

177 views

### Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful.
Let $V \subset H ...

**4**

votes

**0**answers

96 views

### Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question:
Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$:
$$
\frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t),
$$
with smooth initial data ...

**3**

votes

**1**answer

198 views

### Two equivalent definitions of weak solution to parabolic PDE; don't understand proof

(Crossposted from MSE due to no replies)
I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the ...

**3**

votes

**1**answer

192 views

### Heat Equation on $[0,T] \times \mathbb{R}^n$

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.
Do you ...

**0**

votes

**1**answer

82 views

### approximating functions pointwise [closed]

If we have in certain norm
1). $g_j(x) \rightarrow h(x), j\rightarrow\infty$ and
2). $ f_{ij}(x)\rightarrow g_j(x), i\rightarrow \infty, \forall j$ ,
then we can choose a subsequence ...

**5**

votes

**3**answers

359 views

### PDEs involving measures; where to begin?

If I want to learn about existence of weak solutions to PDEs of the form
$$u_t + Au = f$$
or
$$Au = f$$
where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for ...

**2**

votes

**0**answers

72 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 ...

**10**

votes

**4**answers

428 views

### Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...

**0**

votes

**0**answers

81 views

### Interpretation of this PDE solving method needed

I am working on an open source project that deals with the usage of lie groups to solve first order differential equations. For my reference, I am using this paper by Maple (It is there on this ...

**3**

votes

**0**answers

129 views

### How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded

With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider
$$u' + Au = f$$
in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...

**3**

votes

**0**answers

140 views

### Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II:
Assume $u$ is a convex ...

**2**

votes

**2**answers

248 views

### worst regularity of f ensuring u is locally Lipschitz for $\Delta u = f$

Assume $M$ is a Riemannian manifold, and $\Omega $ is a bounded domain. Consider the Poisson equation:
$$\Delta u = f \qquad \text{with }u \in {W^{1,2}}$$
What is the worst regularity of $f$ which ...

**3**

votes

**1**answer

156 views

### A closed extension of the Laplace operator with respect to the supremum norm

Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...

**10**

votes

**0**answers

335 views

### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

**3**

votes

**0**answers

112 views

### A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...

**3**

votes

**2**answers

232 views

### 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 ...

**0**

votes

**0**answers

49 views

### partial maximum principle for elliptic differential operators

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...

**5**

votes

**1**answer

461 views

### Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one.
$I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...

**3**

votes

**1**answer

167 views

### Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...

**3**

votes

**1**answer

97 views

### On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument

Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...

**0**

votes

**0**answers

69 views

### Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?

Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces.
I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that
$$\langle u'(t), ...

**1**

vote

**0**answers

112 views

### Galerkin approximations for parabolic PDE weak solution, getting a uniform bound

(As usual $V \subset H$ are separable Hilbert spaces)
In a book I read this about existence of the solutions to parabolic PDEs:
the approximate solution $u_n(t)$ solves the equation
$$(u_n', ...

**2**

votes

**1**answer

89 views

### Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?

Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow
$$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = ...

**2**

votes

**0**answers

58 views

### Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...

**2**

votes

**0**answers

109 views

### Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...

**2**

votes

**1**answer

239 views

### Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$,
$$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...

**2**

votes

**0**answers

106 views

### linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator:
$$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$
where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...

**1**

vote

**1**answer

104 views

### metric scaling for an inequality

I read a lemma 1.12 in Tobias H.Colding's paper "Ricci curvature and volume convergence"."Suppose that $Ri{c_{{M^n}}} \ge \left( {n - 1} \right)\Lambda {R^{ - 2}}$,p and $q \in M$ with d(p,q)>8R,and ...

**1**

vote

**0**answers

263 views

### A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
...

**2**

votes

**0**answers

91 views

### Regularity of solution of nonlinear equation

Hi!
Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball
$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.
Let $f\in C^{0,\alpha}(B_1)$ ...

**0**

votes

**0**answers

133 views

### Definitions of weak solution to parabolic PDE

Suppose $$u'(t) + A(t)u = f(t)$$ holds in the sense of $L^2(0,T;V')$.
Is the problem
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), ...

**1**

vote

**1**answer

122 views

### Reference for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$

If it is true, where may I find a reference/proof for:
$C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$
where $H$ is a Hilbert space.
Thanks