15

The answer is yes if $\epsilon<1$, and no when $\epsilon\geq 1$.
This follows from Carleman's quasianalyticity criterion, see for example, Hormander, Analysis of linear partial differential operators, Vol. I, Chap I, Section 1, Theorem 1.3.8.
(Carleman's original proof used Complex Analysis, and it was reproduced in the books on the subject. Hormander has ...

answered Sep 21 at 2:57

Alexandre Eremenko

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5

For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as distributions that would be acting on smooth functions) is very well developed in
Bär, Christian; Ginoux, Nicolas; Pfäffle, Frank, Wave equations on Lorentzian ...

4

Yes, they are equivalent. Up to a constant missing and a sign error in the displayed formula, it should read:
$$ (-\Delta_g)^s f(x) = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_g} f(x) - f(x)) t^{-1-s} dt . $$
In fact, this is true for quite general operators. If $L^2$ theory is what you are after, this is a direct consequence of the spectral theorem ...

4

It is not true that this equation always has no solutions in the supercritical case $p > \frac{N + 2}{N - 2}$.
The simplest counterexample is on an annulus, say $\Omega = B_R \setminus B_1$: in this case one may search for radial solutions by separating variables, which reduces to solving the second-order ODE
$$
-u'' - \frac{N - 1}{r} u' = u |u|^{p - 1}
$$...

3

I think that you should read L. Gaarding's seminal paper Linear hyperbolic partial differential equations with constant coefficients, Acta Math 85:1-62 (1951). It explains why hyperbolicity is the appropriate condition for the $C^\infty$-well-posedness of the Cauchy problem.

3

If true, this would follow from the integral expression for the fractional Laplacian:
$$(-\Delta)^{s/2} f(x) = \int_{-\pi}^\pi (f(x) - f(y)) \nu(x - y) dy$$
for an appropriate kernel $\nu$. But, unfortunately, the claimed inequality is false: if, for example, $f$ is a non-zero odd function, then
$$(-\Delta)^{s/2} f(0) = 0$$
(by symmetry), while
$$(-\Delta)^{...

3

There is a "standard" bootstrap argument which can be used to show regularity for semilinear equations. I sketch it here under the assumption that $|f(x, u)| \leq C(1 + |u|^p)$ for some $0 < p < \frac{2n}{n-2}$ in $n \geq 2$ (I know the question was posed in $n = 2$ but it is useful to see the role played by the critical exponent).
Assume we ...

3

This is clearly false as stated, since a necessary condition is that $g(x)\to g(\xi)$ as $x\to\xi$ a. e. in the sphere, but if $g$ is merely $L^2$, this may well fail for every point.
The convergence holds in a weaker sense. For example, it is true that if $g_\rho(x)=g(\rho x)$ for $x\in\partial B_r(0)$ and $\rho<1$, then $g_\rho\to g$ a. e. and in $L^2$ ...

2

Let me write out the equation $J''(u)w = g$ for $g\in H^{-1}$ and $w\in H^1_0(\Omega)$. This is equivalent to
$$
\int_\Omega \nabla w \cdot \nabla v - f''(u)wv = g(v) \quad \forall v\in H^1_0(\Omega).
$$
This is the weak formulation of
$$
-\Delta w - f''(u)w = g$$
plus boundary conditions. To show existence of solutions, you need some assumptions on $f''$ to ...

1

The definite reference for this is the monograph by John Lee "Fredholm operators and Einstein metrics on conformally compact manifolds", Mem. Am. Math. Soc. Series Profile 864, 83 p. (2006), that you can also find on the arXiv: https://arxiv.org/abs/math/0105046
This is limited to "natural" differential operators but can be easily ...

1

The answer is yes: take any smooth function $g_0$ on $\partial D_1$ and solve the Dirichlet problem
$$
\begin{cases}
\Delta g = 0 & \text{ on } D_1\\
g = g_0 & \text{ on } \partial D_1.
\end{cases}
$$
Now extend $g$ to a smooth function on $\mathbb{R}^n$. Multiply by a smooth cutoff function $\eta$ which is $1$ on $D_1$ and compactly supported on $...

1

This paper concerns with your problem.

reference-request fa.functional-analysis ap.analysis-of-pdes ca.classical-analysis-and-odes elliptic-pde

1

This question is difficult to answer because the obstruction to continuous families of solutions is (usually) not technical but substantive, and has to do with the local uniqueness of solutions to the equation for a given $t$.
To see why, consider a simple example which illustrates the basic approach one might take: $F = 0$, $L_t(u) = \text{div} (A_t(x) \...

1

I do not there is a strong relation between the two notions in the general case. Curvature is obviously a local object. On the other hand, the behaviour of the first Dirichlet eigenfunction near a boundary point $p$ is highly non-local.
For example, if $\Omega$ is the union of two balls of different radii connected by a narrow channel, then the normal ...

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