In my recent research, I have investigated the translating soliton equation. This equation appears in the singularity theory of the mean curvature flow of Huisken(1) and Ilmanen(2). A translating soliton is a surface $\Sigma\subset {\mathbb R}^3$ that is a solution of the mean curvature flow when $\Sigma$ evolves purely by translations along some direction $\vec{a}\in {\mathbb R}^3\setminus\{0\}$. In other words, $\Sigma$ is a translating soliton if $\Sigma+t\vec{a}$, $t\in R$, satisfies that fixed $t$, the normal component of the velocity vector $\vec{a}$ at each point is equal to the mean curvature at that point. For the initial surface $\Sigma$, this implies that
$$2H=\langle N,\vec{a}\rangle\quad (1),$$ where $N$ is the Gauss map of $\Sigma$. Translating solitons appear in the singularity theory of the mean curvature flow. After scaling, near type II-singularity points on the surfaces evolved by mean curvature vector, Huisken, Sinestrari and White(3) demonstrated that the limit flow with initial convex surface is a convex translating soliton. The interest on translating solitons have grown in the recents years and the literature is great.
After a change of coordinates, we suppose $\vec{a}=(0,0,1)$. In a non-parametric expression, equation (1) writes as
$$2H=\langle N,\vec{a}\rangle\quad (1),$$ where $N$ is the Gauss map of $\Sigma$. Translating solitons appear in the singularity theory of the mean curvature flow. After scaling, near type II-singularity points on the surfaces evolved by mean curvature vector, Huisken, Sinestrari and White(3) demonstrated that the limit flow with initial convex surface is a convex translating soliton. The interest on translating solitons have grown in the recents years and the literature is great.
After a change of coordinates, we suppose $\vec{a}=(0,0,1)$. In a non-parametric expression, equation (1) writes as
$$\mbox{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)=\frac{1}{\sqrt{1+|Du|^2}} \quad (2)$$
in a smooth domain $\Omega\subset{\mathbb R}^2$, where $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$. If $(x,y,z)$ are the canonical coordinates in Euclidean space $({\mathbb R}^3,\langle,\rangle)$ and $\Sigma_u=\{(x,y,u(x,y)): (x,y)\in\Omega\}$ is the graph of the function $u$, the left-hand side of (2) is twice the mean curvature $H$ of $\Sigma_u$ at each point $(x,y,u(x,y))$. In other words, a surface in Euclidean space satisfies locally the translating soliton equation if and only if the mean curvature at each point is the half of the cosine of the angle that makes $N$ with the vertical direction $\vec{a}=(0,0,1)$.
My first contact with equation (2) was around 2000 when I began with the study of the Dirichlet problem of the constant mean curvature equation(4). I had to read the classical article of Serrin(5) on the existence of solutions of the Dirichlet problem. This article is huge! More than 80 pages! so much for me. Serrin gave a systematic treatment of the Dirichlet problem for a large class of quasilinear non-uniformly second order elliptic equations. Following the Leray-Schauder fixed point theorem and Hölder estimates theory of Ladyzenskaja and Ural'ceva, Serrin establishes the necessary and sufficient conditions for the solvability of the Dirichlet problem for arbitrary boundary data. Possibly, the most known result of this paper is the case of the constant mean curvature equation, $2H=c$, $c$ is a constant. In such a case, the Dirichlet problem has a solution for arbitrary smooth boundary data $\varphi$ if and only if the curvature $\kappa$ of $\partial\Omega$ with respect to the inward normal direction satisfies $\kappa\geq 2|H|$. If the solution exists, it is unique.
However, the article (5) covers many other types of quasilinear elliptic equations and this is the situation of the translating soliton equation. And le voilà, equation (2) was there! Exactly in pages 477-478, Serrin considers two families of quasilinear elliptic equations and one of them coincides with (2). The equation (96) of (5) is
$$ (1+q^2)r-2pqs+(1+p^2)t=2H(1+p^2+q^2)^{n/2} \quad (3)$$
where $H$ and $n$ are two real constants. In $n=2$, equation (3) coincides with (2) (assuming $2H=1$). Notice that if $n=3$, the expression (3) is the constant mean curvature equation.
Now, after almost two decades, I see the interest of this equation, but I think that it would be good to give the credits to Serrin. Not only him. Now, and preparing a paper about this equation, I returned to read the Serrin's paper and there is a previous literature about the translating soliton equation! It is due to Bernstein, but that's the story for the following entry.
References
My first contact with equation (2) was around 2000 when I began with the study of the Dirichlet problem of the constant mean curvature equation(4). I had to read the classical article of Serrin(5) on the existence of solutions of the Dirichlet problem. This article is huge! More than 80 pages! so much for me. Serrin gave a systematic treatment of the Dirichlet problem for a large class of quasilinear non-uniformly second order elliptic equations. Following the Leray-Schauder fixed point theorem and Hölder estimates theory of Ladyzenskaja and Ural'ceva, Serrin establishes the necessary and sufficient conditions for the solvability of the Dirichlet problem for arbitrary boundary data. Possibly, the most known result of this paper is the case of the constant mean curvature equation, $2H=c$, $c$ is a constant. In such a case, the Dirichlet problem has a solution for arbitrary smooth boundary data $\varphi$ if and only if the curvature $\kappa$ of $\partial\Omega$ with respect to the inward normal direction satisfies $\kappa\geq 2|H|$. If the solution exists, it is unique.
However, the article (5) covers many other types of quasilinear elliptic equations and this is the situation of the translating soliton equation. And le voilà, equation (2) was there! Exactly in pages 477-478, Serrin considers two families of quasilinear elliptic equations and one of them coincides with (2). The equation (96) of (5) is
$$ (1+q^2)r-2pqs+(1+p^2)t=2H(1+p^2+q^2)^{n/2} \quad (3)$$
where $H$ and $n$ are two real constants. In $n=2$, equation (3) coincides with (2) (assuming $2H=1$). Notice that if $n=3$, the expression (3) is the constant mean curvature equation.
Now, after almost two decades, I see the interest of this equation, but I think that it would be good to give the credits to Serrin. Not only him. Now, and preparing a paper about this equation, I returned to read the Serrin's paper and there is a previous literature about the translating soliton equation! It is due to Bernstein, but that's the story for the following entry.
References
[1] Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45-70 (1999)
[2] Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 108, x+90 (1994)
[3] White, B.: Subsequent singularities in mean-convex mean curvature flow. Calc. Var. 54, 1457-1468 (2015)
[4] López, R.: Constant mean curvature graphs on unbounded convex domains. J. Diff. Equations, 171, 54-62 (2001)
[5] Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Phil. Trans. R. Soc. Lond. 264, 413-496 (1969)
