In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1norm of potentials is known. We establish the minimization character-ization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough.

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Let$G$denote a totally disconnected locally compact metric abelian group with translation invariant metric$d$and character group$Γ$_{$G$}. The Lipschitz spaces are defined by $$Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$$ where$τ$_{$a$}$f$:$x$→$f$($x-a$) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α;$p$)⊂$L$^{$r$}($Γ$_{$G$}), where α>1/$p$+1/$r$−1≧0 and 1≦$p$≦2. In the case$G$is compact the corresponding result holds for α>1/$r$−1/2 and$p$>2. In addition for$G$non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

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In this paper, we give an integral approximation for the elliptic operators with anisotropic coecients on smooth manifold. Using the integral approximation, the elliptic equation is tranformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.