Gamma-convergence and the least squares method
The authors study the limits (as \(h\to +\infty\)) of equations of the form \[ u_ t- a_ h(y) u_ x= 0\qquad (t,x,y)\in \Omega\subset \mathbb{R}^ 3 \] through the analysis of the associated functionals \[ F_ h(u, \Omega)= \int_ \Omega | u_ t- a_ h(y) u_ x|^ 2 dt dx dy. \] When the coefficients \(a_ h\) are only weakly converging, the solutions \(u_ h\) are not compact in \(L^ 2\), so that the natural candidate for the limit functional \(F\) is the \(\Gamma\)-limit of \(F_ h\) in the weak \(L^ 2(\Omega)\) topology. This limit functional is represented by using the Young limit of the coefficients \(a_ h(y)\), which is a family \(\nu_ y\) of probability measures indexed by \(y\). In this way the functional \(F\) is given by the inf-convolution \[ \begin{multlined} F(u)= \int_ I dy\Bigl[\inf\Bigl\{\int_ J \int_{\Omega_ y}| w_ t- jw_ x|^ 2 dt dx d\nu_ y(j):\\ w\in L^ 2(J,\nu_ y; L^ 2(\mathbb{R}^ 2)),\quad\int_ J w d\nu_ y(j)= u(\cdot, y)\Bigl\}\Bigl],\end{multlined} \] where \(I\) is the projection of \(\Omega\) on the \(y\) axis, \(J\subset \mathbb{R}\) is a compact interval containing all values of \(a_ h\), and \(\Omega_ y= \{(t,x): (t,x, y)\in \Omega\}\) is the \(y\)-slice of \(\Omega\). For instance, when \(a_ h\) are given by \[ a_ h(y)= \begin{cases} +1 &\text{if } [hy]\text{ is even}\\ - 1 &\text{if }[hy]\text{ is odd }\end{cases} \] the functional \(F\) becomes \[ F(u)= 2\inf\left\{ \int_ \Omega ((v_ t+ v_ x)^ 2+ (w_ t- w_ x)^ 2)dt dx dy: v+ w= u\right\}. \] {}.
scientific article; zbMATH DE number 711474
scientific article