Let be an odd function of a class C such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f\left(x\right)/x$ increases on . We approximate the positive solution of Δ = 0, on ${}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ]0,[ with adequate non-homogeneous Dirichlet conditions. We show that the error tends to zero exponentially fast, in the uniform norm.

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a Borelian function and consider the following problems $$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$
 $$inf\left\{F^{**}\left(y\right)={\int }_a^b{L}^{**}(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}·\left(P^{**}\right)$$ We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ if is just continuous in . We then extend a result of Cellina on the Lipschitz regularity of the minima of when is not superlinear.

We investigate the value function of the Bolza problem of the Calculus of Variations
$$V(t,x)=inf\left\{{\int }_0^{t}L(y\left(s\right),y^{\text{'}}\left(s\right))ds+\varphi \left(y\left(t\right)\right):y\in W^{1,1}(0,t;{ℝ}^n),y\left(0\right)=x\right\},$$ with a lower semicontinuous Lagrangian and a final cost $\varphi$, and show that it is locally Lipschitz for whenever is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize...

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand : × $\to$ $\cup$ $\left\{\infty \right\}$, where is the -dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

We investigate the minima of functionals of the form $${\int }_{[a,b]}g\left(\dot{u}\left(s\right)\right)\mathrm{d}s$$ where is strictly convex. The admissible functions $u:[a,b]⟶ℝ$ are not necessarily convex and satisfy $u\le f$ on , , , is a fixed function on . We show that the minimum is attained by $\overline{f}$, the convex envelope of .