Asymptotic flatness of Morrey extremals

Abstract

We study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions u for Morrey’s inequality on $\mathbb {R}^n$. In particular, we compute the limit of u(x) as $|x|\rightarrow \infty$ and show |x||Du(x)| tends to 0. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-\Delta _pu=c(\delta _{x_0}-\delta _{y_0})$ for some $c\in \mathbb {R}$ and distinct $x_0,y_0\in \mathbb {R}^n$. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded p-harmonic functions on exterior domains of $\mathbb {R}^n$ for $p>n$.

Appendix A: Numerical method

We will now discuss the method used to approximate the solution of PDE (1.3) displayed in Fig. 1. It turns out that this method also can be adapted to obtain approximations of solutions of the multipole Eq. (5.4), as exhibited in Figs. 3 and 4. For simplicity, we will focus on the particular case of approximating a solution u of the PDE

$$\begin{aligned} -\Delta _pu=\delta _{(0,1)}-\delta _{(0,-1)} \end{aligned}$$

(A.1)

in $\mathbb {R}^2$. We will also change notation and use ordered pairs (x, y) to denote points in $\mathbb {R}^2$ so that $u=u(x,y)$.

Observe that any solution $u\in {{\mathcal {D}}}^{1,p}(\mathbb {R}^2)$ of (A.1) minimizes

$$\begin{aligned} \iint _{\mathbb {R}^2}\frac{1}{p}|Dv|^pdxdy- (v(0,1)-v(0,-1)) \end{aligned}$$

(A.2)

among all $v\in {{\mathcal {D}}}^{1,p}(\mathbb {R}^2)$. For a given $\ell \in \mathbb {N}$, we may also consider the problem of minimizing

$$\begin{aligned} \int ^\ell _{-\ell }\int ^\ell _{-\ell }\frac{1}{p}|Dv|^pdxdy- (v(0,1)-v(0,-1)) \end{aligned}$$

amongst $v\in W^{1,p}([-\ell ,\ell ]^2)$. It is not hard to show this problem has a minimizer $u_\ell \in W^{1,p}([-\ell ,\ell ]^2)$. Further, it is routine to check that $u_\ell (x,y)-u_\ell (0,0)$ converges to u(x, y) for each $(x,y)\in \mathbb {R}^2$ as $\ell \rightarrow \infty$, where u is the unique minimizer of (A.2) with $u(0,0)=0$. Consequently, we will focus on approximating $u_\ell$.

Below we will show how to derive a discrete version of our minimization problem for $u_\ell$. Then we will explain how to use an iteration scheme based on a quasi-Newton method. Again we emphasize that all of the figures in this article were based on this method or minor variants to account for differences in the particular examples we considered.

1.1 Appendix A.1: Discrete energy

Let us fix $\ell \in \mathbb {N}$ ($\ell \ge 2$) and discretize the interval $[-\ell ,\ell ]$ along the x-axis with

$$\begin{aligned} x_i=-\ell +(i-1)h \end{aligned}$$

for $i=1,\dots , M$. Here

$$\begin{aligned} h=\frac{2\ell }{M-1}, \end{aligned}$$

and we note that each of the subintervals $[x_1,x_{2}],\dots , [x_{M-1},x_{M}]$ has length h. We can do the same for the interval $[-\ell ,\ell ]$ along the y-axis and obtain

$$\begin{aligned} y_j=-\ell +(j-1)h \end{aligned}$$

for $j=1,\dots , M$. Our goal is to derive an appropriate energy specified for functions defined on the grid points $(x_i,y_j)$.

To this end, we observe that if $v:[-\ell ,\ell ]^2\rightarrow \mathbb {R}$ is sufficiently smooth

$$\begin{aligned}&\int ^\ell _{-\ell }\int ^\ell _{-\ell }|Dv|^pdxdy\\&\quad \approx \sum ^{M-1}_{i,j=1}|Dv(x_i,y_j)|^ph^2\\&\quad =\sum ^{M-1}_{i,j=1}\left( v_{x}(x_i,y_j)^2+v_{y}(x_i,y_j)\right) ^{p/2}h^2\\&\quad \approx \sum ^{M-1}_{i,j=1}\left( \left( \frac{v(x_i+h,y_j)-v(x_i,y_i)}{h}\right) ^2+\left( \frac{v(x_i,y_j+h)-v(x_i,y_i)}{h}\right) ^2\right) ^{p/2}h^2\\&\quad =\sum ^{M-1}_{i,j=1}\left( \left( \frac{v(x_{i+1},y_j)-v(x_i,y_i)}{h}\right) ^2+\left( \frac{v(x_i,y_{j+1})-v(x_i,y_i)}{h}\right) ^2\right) ^{p/2}h^2\\&\quad = h^{2-p}\sum ^{M-1}_{i,j=1}\left( \left( v(x_{i+1},y_j)-v(x_i,y_i)\right) ^2+\left( v(x_i,y_{j+1})-v(x_i,y_i\right) ^2\right) ^{p/2}. \end{aligned}$$

We also suppose $h=1/k$ for some $k\in \mathbb {N}$ which gives

$$\begin{aligned} M=2\ell k+1 \end{aligned}$$

and

$$\begin{aligned} (x_{\ell k+1}, y_{(\ell +1) k+1})=(0,1)\quad \text {and}\quad (x_{\ell k+1}, y_{(\ell -1) k+1})=(0,-1). \end{aligned}$$

As a result, we will attempt to minimize

$$\begin{aligned} E(v)= & {} \frac{1}{p}k^{p-2}\sum ^{M-1}_{i,j=1}\left( \left( v_{i+1,j}-v_{i,j}\right) ^2+\left( v_{i,j+1}-v_{i,j}\right) ^2\right) ^{p/2}\nonumber \\&-(v_{\ell k+1,(\ell +1)k+1}-v_{\ell k+1,(\ell -1)k+1}) \end{aligned}$$

(A.3)

over the $M^2-1$ variables

$$\begin{aligned} v= \left( \begin{array}{ccccc} v_{1,1} &{} v_{1,2}&{}\dots &{} v_{1,M-1}&{} v_{1,M}\\ v_{2,1} &{} v_{2,2}&{}\dots &{} v_{2,M-1}&{} v_{2,M}\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ v_{M-1,1} &{} v_{M-1,2} &{} \dots &{} v_{M-1,M-1} &{}v_{M-1,M} \\ v_{M,1} &{} v_{M,2} &{} \dots &{} v_{M,M-1} &{} \end{array}\right) . \end{aligned}$$

A minimizer $v=(v_{ij})$ for E would then form an approximation for $u_\ell$ on the grid points $(x_i,y_j)$

$$\begin{aligned} u_\ell (x_i,y_j)\approx v_{ij}. \end{aligned}$$

1.2 Appendix A.2: Quasi-Newton method

We used a multidimensional version of the secant method to approximate minimizers of the discrete energy E defined above in (A.3). In particular, since E is convex we only need to approximate a $v=(v_{ij})$ such that

$$\begin{aligned} \partial _{v_{ij}} E(v)=0 \end{aligned}$$

for each $i,j=1,\dots , M$ with $(i,j)\ne (M,M)$.

First we chose the initial guesses

$$\begin{aligned} v^0_{ij}=0 \end{aligned}$$

and

$$\begin{aligned} v^1_{ij}=g(x_i,y_j). \end{aligned}$$

Here

$$\begin{aligned} g(x,y)=-\frac{1}{4\pi }\log \left[ \frac{x^2+(y-1)^2+10^{-2}}{x^2+(y+1)^2+10^{-2}}\right] \end{aligned}$$

is approximately equal to

$$\begin{aligned} g_0(x,y)=-\frac{1}{4\pi }\log \left[ \frac{x^2+(y-1)^2}{x^2+(y+1)^2}\right] , \end{aligned}$$

which is a solution of the Dipole equation $-\Delta g_0=\delta _{(0,1)}-\delta _{(0,-1)}$ in $\mathbb {R}^2$.

Then we performed the iteration

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle v^{m+1}_{ij}=v^{m}_{ij}-\tau _m\partial _{v_{ij}} E(v^m) \\ \\ \tau _m:=\frac{\displaystyle \sum \nolimits _{ij}(v^m_{ij}-v^{m-1}_{ij}) (\partial _{v_{ij}} E(v^m)-\partial _{v_{ij}} E(v^{m-1}))}{\displaystyle \sum \nolimits _{ij}\left( \partial _{v_{ij}} E(v^m)-\partial _{v_{ij}} E(v^{m-1})\right) ^2} \end{array}\right. } \end{aligned}$$

for $m=1,2,3,\dots$ until the stopping criterion

$$\begin{aligned} \max _{ij}\left| \partial _{v_{ij}} E(v^m)\right| <10^{-6} \end{aligned}$$

was achieved. The iteration was computed for all $i,j=1,\dots , M$ except for $(i,j)\ne (M,M)$.