Class Of Quasilinear Elliptic Equations Research Articles

On Pointwise Estimates for a Finite Element Solution of Nonlinear Boundary Value Problems

The solution of the Dirichlet problem for a class of quasilinear elliptic equations is approximated by the simplest finite element method. Piecewise linear trial functions are used over a regular triangulation of the n-dimensional domain with simplices of diameter at most h. Under conditions which are sufficient for the existence of a classical solution we prove pointwise error estimates. In the uniformly elliptic case and for the minimal surface problem the finite element solution converges uniformly with a rate which depends on n and the shape of the domain. It is, e.g., at least $h^{{3 / 2}} | {\log h} |^{{1 / 2}} $ for $n = 2$ and h for $n = 3$. Numerical results are given for two minimal surfaces.

On the behavior of solutions of quasi-linear elliptic equations

Recently it has been noted that the minimal surface equation, and other related quasi-linear elliptic equations have the property that the boundary values of a solution on only part of its domain, may impose bounds upon the solution of all points of its domain. A result of this type first appeared in [6] as a consequence of the fact that the minimal surface equation possesses a solution; e.g., the minimal surface of Scherk; which becomes positively infinite on a straight boundary segment. Finn [4] has obtained stronger results of this type, by showing that if D is a domain, bounded in part by an arc F; and if q is a solution of an elliptic equation in D whose gradient becomes infinite as F is approached; then 4 majorizes in D, any solution which it majorizes on OD -1F. In particular, the catenoid b0(r) = - a cos h lr/a is a solution of the minimal surface equation in r > a; and limr a O0b0 / Or = - 00, while limr-a4o(r) is finite. It follows therefore that O0(r) majorizes in a < r < b, any solution which is < X0(b) on r = b, and hence any such solution is uniformly bounded in a < r < b. By applying the above argument to a solution defined in the punctured disc, and letting a -+0, Finn obtained an elegant new proof of the removability of isolated singularities of solutions of the minimal surface equation. These results extend immediately to the class of radially symmetric variational problems in n-variables, whose radially symmetric solutions have the essential properties of the catenoid. This class was characterized by Finn in [4]. Extensions to a wide class of quasi-linear elliptic equations in 2 variables was given by the author, in reference [1], by constructing catenoid-like super-solutions. Using the catenoid, or catenoid-like solutions or super-solutions as comparison functions the argument of Finn leads to the following theorem, which is valid for the class of equations possessing such solutions or super-solutions: Let D be a domain lying exterior to a circle, and bounded in part, by an arc F of the circle. Then there is a uniform bound on 4 in Do which depends only upon the supremum of 4 on OD - r.