**2**

votes

**1**answer

195 views

### Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ...

**3**

votes

**0**answers

185 views

### W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...

**2**

votes

**1**answer

276 views

### Fully non-linear PDE

A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...

**1**

vote

**1**answer

134 views

### radial limits of subharmonic functions

Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every ...

**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) = ...

**1**

vote

**1**answer

233 views

### Blow up of solutions to parabolic PDEs

I am looking for a text or answer detailing the blowup of solutions to parabolic PDE (eg. heat equation) in Sobolev space setting. I heard blowup is related to size of domain but I can't find any nice ...

**2**

votes

**2**answers

150 views

### Changing the space of test functions in PDEs

Let $V \subset H \subset V'$ be a Hilbert triple.
We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying
$$\int_0^T u(t)\varphi'(t)=-\int_0^T ...

**1**

vote

**1**answer

225 views

### What are the advantage of using operational calculus for numerically solving pde compared to FE or FD?

For numerically solving a partial differential equation (PDE) what advantage does operational calculus (OC) has over common methods like finite difference (FD), and finite element (FE)?
I mean OC in ...

**3**

votes

**1**answer

204 views

### Boundary flux maximizing drift (velocity) vector fields for 2D heat equation

Looking for literature / known results on the following class of problems:
Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar ...

**3**

votes

**1**answer

220 views

### Null sets in PDE

Consider the weak formulation: 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), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = ...

**4**

votes

**1**answer

139 views

### diffusions corresponding to estimators

I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...

**15**

votes

**5**answers

2k views

### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**3**

votes

**2**answers

384 views

### Iwaniec's conjecture

Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...

**5**

votes

**3**answers

621 views

### Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...

**2**

votes

**1**answer

170 views

### Does a particular iteration produce a weak solution to a non linear pde?

Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...

**2**

votes

**1**answer

431 views

### Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...

**6**

votes

**1**answer

134 views

### Ergodic Mean for Schrodinger flow

Let us consider the linear Schrödinger equation in $\mathbb{R}^N$
$$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$
with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its ...

**5**

votes

**0**answers

173 views

### Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: ...

**1**

vote

**1**answer

198 views

### Nonlinear parabolic PDEs existence with Galerkin method?

Can someone give me some references to read where existence/uniqueness of nonlinear parabolic PDE are treated via the Galerkin method or fixed point methods or something like that (anything but ...

**10**

votes

**2**answers

597 views

### the spectrum of the Laplacian and Dirac operator on $S^3$

A paper on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$:
The eigenvalues of the vector Laplacian on divergenceless vector
fields is $(\ell + 1)^2$ ...

**4**

votes

**1**answer

201 views

### Local boundedness of weak solutions of heat equations…?

(Please, what is going on?) The following claim is from a paper which was apparently reviewed by László Erdös, Zhongwei Shen, and Bernard Heffler. Someone tell me it's true. Surely it's true. The ...

**1**

vote

**2**answers

100 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

99 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

148 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 ...

**4**

votes

**1**answer

290 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

138 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

384 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

273 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

133 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 ...

**8**

votes

**1**answer

200 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

318 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

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$, ...

**4**

votes

**2**answers

407 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

418 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

149 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

124 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

187 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

81 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

109 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

204 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

128 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

241 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

221 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

85 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

419 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

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 ...

**11**

votes

**4**answers

716 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 ...

**3**

votes

**0**answers

148 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

202 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 ...