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Too long for a comment. I would say that your problem is a semi-global solvability question. You will find in chapter 26 of Hörmander’s ALPDO a precise definition for that property which suits well th …

Your functions $u_j$ are solutions of a semi-linear elliptic equation and, for $p>2$ the function $t\mapsto\vert t\vert^{p-1}t=\phi(t)$ is $C^2$; as a consequence, each $u_j$ is $C^\alpha$ with some p …

There exist $c>0$ and $s>0$, such that for all smooth functions $v$ compactly supported in $\Omega$, $$ \sum_{1\le j\le m}\Vert X_jv\Vert_{L^2}\ge c\Vert v\Vert_{W^{s,2}}. $$ The largest (i.e. the bes …

You can always consider the second (distribution) derivative of a continuous function and require that it is non-negative, which means that for all $T\in \mathbb R^n$ and all $\phi\in \mathscr D(\Omeg …

Let us set $L_1=\sum_{1\le j\le m}X_j^2$, where the $X_j$ are smooth real vector fields satisfying Hörmander's condition so that $L_1$ is subelliptic with an estimate $$ \Vert L_1 u\Vert_{H^s}\ge C\Ve …

Yes you do have a generalization of your elliptic inequality to the $L^p$ case for $p\in (1,+\infty)$. In fact the operators with symbols in the class $S^0_{1,0}$ (as in your question) are bounded on …

Yes, it is a classical pseudo-differential operator of order $-1$ with principal symbol $\vert p_D(x,\xi)\vert^{-1}$ where $p_D$ is the principal symbol of $D$; it is also possible to prove that you h …

The operator $(-∆)^s$ is the Fourier multiplier $\vert\xi\vert^{2s}$ and thus deserves to be denoted by $\vert D\vert^{2s}$. Now the distribution $u_\alpha(x)=x_+^\alpha$ is homogeneous with degree $\ …

A semi-linear PDE reads $ \mathcal Lu=F(u), $ where $\mathcal L$ is a linear operator and $F$ is a function. A quasi-linear PDE with order $m$ reads $ \mathcal L\bigl((\partial_x^\alpha u)_{\vert \alp …

No. The operator $(-∆)^s$ is the Fourier multiplier $\vert \xi\vert^{2s}$ so that, say for $f$ in the Schwartz space whose Fourier transform vanishes near the origin, we have $$ \Vert(-∆)^{-s} f\Vert …

If I understand your question correctly, you speak about interior regularity. Let me quote a classical result for linear elliptic equations with $C^\infty$ coefficients, even true for pseudo-different …

First a classical result about elliptic operators: in dimension $\ge 3$ the order of elliptic operators is even. In dimension 2, say in $\mathbb R^2$ you have elliptic vector fields such as $$ \bar \p …

Let me start with a constant coefficient operator $$ P(D)=\sum_{1\le j, k\le n} a_{jk} D_jD_k,\quad D_j=\frac{\partial }{i\partial x_j}. $$ Note that in two dimensions, you have elliptic operators wit …

I guess that your question is ill-formulated: you have to respect the homogeneity on both sides of the inequality. Let us say that for $f$ in the Schwartz space you always have $$ \Vert f\Vert_{W^{t,q …

About your second question. If I understand things correctly, you want to solve the Dirichlet problem $$ \Delta u = \operatorname{div} F, \quad u_{\vert \partial \Omega}=0. $$ Since your open set is s …