Fully Nonlinear Equations with Applications to Grad Equations in Plasma Physics

We will consider $W^{2,p}$-viscosity solutions $u:\Omega\subset\mathds{R}^{n}\longrightarrow\mathds{R}$ for

where $\Omega\subset\mathds{R}^{n}$ is an open, bounded and connected set, with $C^{1,1}$ boundary. The operator $F:\mathcal{S}\longrightarrow\mathds{R}$, is a convex, uniformly elliptic operator with ellipticity constants $0<\lambda\leq\Lambda$, where $\mathcal{S}:=\{$real n$\times$n symmetric matrices$\}$. For simplicity we will assume $F(0)=0$. We will also require that $F$ satisfies the following structure condition

for all $M,N\in\mathcal{S}$. Here, $\mathcal{M}^{-}$ and $\mathcal{M}^{+}$ are the extremal Pucci operators

and

where $e_{i}=e_{i}(M)$ are the eigenvalues of $M$. In the right hand side of (1), $|\cdot|$ denotes the n-dimensional Lebesgue measure, and $g:[0;|\Omega|]\longrightarrow\mathds{R}$ is a continuous function. We will adopt the notation

for the superlevel sets of $u$. Finally, we consider a boundary value $\psi\in W^{2,p}(\Omega)$ for $p>n$.

The motivation to study this problem is to generalize Grad equations in plasma physics, and its approximations, to nonlinear operators. These equations were introduced by Harold Grad in [4], and appear in the literature as Queer Differential Equations(QDE), or Grad Equations. They arise in modelling plasma, which is confined under magnetic forces in a toroidal container. Grad noticed that a simplified version of plasma equations was possible using $u^{*}$, the increasing rearrangement of $u$:

Here is where we start building a connection with our approximation problem (1). Notice that heuristically, $u^{*}$ is the inverse of the measure of the sublevel sets of $u$. In [4], he demonstrated that there are profile functions $\mu$ and $\nu$ which are prescribed by the dynamics of the plasma; consequently his equation reads

for some power $\gamma$. For clarity we avoided the arguments: $u$ and its derivatives are evaluated at some point $x$ while the rearrangements and its derivatives are evaluated at $t:=|u<u(x)|$. Many authors attacked the problem trying to approximate these equations. The first one was introduced by Roger Temam in [8], and then improved by Mossino and Temam in [7]. They studied properties of directional derivatives of the rearrangement function, and proved existence results for

Years later, Laurence and Stredulinsky, in [5] and [6], studied a model equation, closer to Grad’s formulation. They considered the particular case when $\gamma=2$, $\mu\equiv 1/2$ and $\nu\equiv 0$ obtaining

Even this simplified case presents many difficulties. The authors introduced a very interesting approach to the problem: they described an approximation with solutions to a $N-$free boundary problem. In order to apply this process they assumed extra regularity for the level sets of a solution, which is mentioned later in Section 3.

The idea behind this paper is the following: all of these previous papers addressed the problem with a variational method for the Laplacian; instead, we will use a viscosity approach for a general family of fully nonlinear operators. A similar equation to the one of Mossino and Temam is studied, and even for the case with the Laplacian we improve the regularity results.

The paper is organized as follows: In the first section we cite some preliminary definitions. Mainly, we state the basics of $W^{2,p}$-viscosity solutions. The classic viscosity solutions’ theory does not apply to this particular problem because of our right hand side in (1). Disregarding the regularity of $g$ and $u$, we notice that having $|u=c|>0$ for some constant $c$ makes the right hand side of our equation discontinuous. Therefore, we adopt this $W^{2,p}$-viscosity notion defined in [2] by Caffarelli, Crandall, Kocan and Swiech which allows merely measurable “ingredients”. In their paper they proved existence and interior $W^{2,p}$-estimates for solutions to an equation with a fixed right hand side $f(x)$. Strongly based on their results, Winter in [9] extended this regularity up to the boundary proving global $W^{2,p}$-estimates for viscosity solutions and an existence result for $W^{2,p}$-strong solutions. For clarity in the presentation, the results from the literature that will be used through the paper will be addressed at the Appendix A.

In section 2 we state and prove the main theorem of existence and global regularity. The idea of the proof is to

• •

  freeze $u$ in the right hand side

• •

  solve the resulting equation using [2] theory

• •

  build a sequence of right hand sides and solutions

• •

  use a fixed point argument and a convergence theorem to find a solution

In section 3 we prove more regularity under additional hypothesis. As long as $|\nabla u|$ is uniformly bounded below, or equivalently, if we have a uniform interior ball condition for the level sets of $u$, then we have $C^{0,\alpha}$ regularity for the right hand side. This estimate turns into $C^{2,\alpha}$ regularity for the solution $u$. We cannot ensure regularity for the level sets, but if we start with a regular enough domain, say $\partial\Omega$ with a uniform interior ball condition, then we gain $C^{2,\alpha}$ regularity for $u$ in a neighborhood of the boundary.

First we are going to present the definitions of viscosity solutions for fully nonlinear equations with measurable ingredients, described in the paper of Caffarelli-Crandall-Kocan-Swiech [2]. In this setting we work with the problem

where our right hand side is a fixed measurable function $f$.

Because we will use Winter’s results, we also add the definition of $W^{2,p}$-strong subsolutions.

In this section we state and prove existence and a first global regularity result.

The structure of the proof for Theorem 2.1 is somehow simple; we set an approximating problem (6), we prove the existence of a solution for it and then we take the limit to obtain the solution to (1). Before presenting this approximating problem in Lemma 2.3, we give a quick explanation on the reasoning behind it. Recall that the results from the appendix will be used next: existence and uniqueness, fixed point, and convergence.

If in (1) we freeze a function $v\in Lip(\Omega)$ for the right hand side, i.e. $f_{v}(x):=g\Big{(}|\{y\in\Omega:v(y)\geq v(x)\}|\Big{)}$, we get

Then the hypothesis of Theorem A.1 are satisfied and there exists a unique $W^{2,p}$-viscosity solution $u$ to (4). The next step would be to apply the fixed point Theorem A.2 for the application $T(v)=u$. The problem is that we cannot ensure continuity for $T$ because of the right hand side of (4). Not even if we require more regularity for $v$(not even $C^{\infty}$ works). We will overcome this inconvenient solving an auxiliary problem with a smoothened right hand side which allows us to perform the fixed point argument. Given $v\in Lip(\Omega)$, $\varepsilon>0$, consider

Because $f_{v}^{\varepsilon}\in L^{p}(\Omega)$, using Theorem A.1 we have existence and uniqueness of a $W^{2,p}$-viscosity solution $u\in W^{2,p}(\Omega)$ to (5) with the estimate

Now we can state our approximation lemma.

The purpose of finding such a $u_{\varepsilon}$, was to approximate a solution for (1). Then the following question is if we can take the limit $\varepsilon\longrightarrow\infty$.

The last remark of this section is that we obtain an explicit formula for the $0$ Dirichlet problem in a ball.

In order to gain more regularity for our solution $u$ we probably need to get some regularity for the right hand side $f_{u}$. So far, in the case when $u$ has flat regions, $f_{u}$ is not even continuous. In principle, this discontinuity does not depend on the regularity of $u$ but on its flat regions. We can prove that under the negativity of $g$, $u$ is not allowed to have these flat regions with positive measure.

Then, for this specific case we obtain continuity for $f_{u}$. But we need at least $C^{0,\alpha}$ regularity on $f_{u}$ in order to apply Schauder type estimates to obtain $u\in C^{2,\alpha}$. We will have this regularity in two particular cases listed in the next two theorems. The first one is an adaptation(simplification) to Laurence and Stredulinsky’s theorem and requires an additional lower bound for the gradient.

In other words, the theorem asserts that if we have an uniform lower bound for the gradient(away from the maximum of $u$), then we get: regularity for the level sets of $u$ and we discard a possible “flatness” which ruins the smoothness of $f_{u}$. The proof presented in [6] includes an approximation argument by $C^{\infty}_{0}$ functions and coarea formula.

This last theorem translates into regularity for our problem. We state this in the following corollary.

The second theorem states that, under certain conditions, a barrier argument implies lower bounds as in Theorem 3.2.