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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 …

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 …

In the first place, you can write $ \mathbf 1_\Omega=\sum_{k\ge 1}\chi_k, \ \chi_k\in C^\infty_c(\Omega). $ Let $\nu\in C^\infty(\mathbb R^n)$, vanishing on $B(0,1/2)$, equal to 1 on $B(0,1)^c$: let u …

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 …

Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel …

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 …

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 …

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 are two different parts in elliptic regularity theory. The first and easier is interior regularity, which can be proven for $p\in (1,+\infty)$ essentially by the same method as for $p=2$, usin …

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 …

Writing $$ \langle-\Delta u + au, u\rangle_{L^2(\Omega)}=\langle f, u\rangle_{L^2(\Omega)}, $$ using the Dirichlet boundary condition, you get $$ \Vert \nabla u\Vert_{L^2(\Omega)}^2+\langle au, u\rang …

The problem is indeed coming form the fact that singular integrals, such as the Hilbert transform, although bounded on $L^p$ for $1<p<+\infty$ are failing to be bounded on $L^1$ or $L^\infty$. Howev …

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 …

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 …

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 …