Extension and applications of the integral inequality of C. Miranda (Q1824760)

The author proves the following existence Theorem (Theorem 4.II): Let \(a_ 1(x),...,a_ n(x)\) be functions defined in the cube \(\Omega =]0,1[^ n\) and such that: \[ a_ i\in W^{1,4+2\alpha}(\Omega)\quad with\quad -1/6\leq \alpha <0,\quad \inf_{\Omega} a_ i(x)>0. \] Then for each \(f\in L^{\infty}(\Omega)\) there exists at least one solution \(u\in W^{2,2+2\alpha}(\Omega)\cap L^{\infty}(\Omega)\) of the equation: \[ (3)\quad \sum^{n}_{i=1}a_ i(x)D^ 2_{x_ i}u(x)=f(x)\quad in\quad \Omega \quad [with\quad u|_{\partial \Omega}=0]. \] The condition \(\alpha <0\) compels the author to establish an extension of an integral inequality of C. Miranda, in which \(\alpha\) is allowed to be negative. The inequality (Theorem 1.I) is: \[ (4)\quad | u'|_{E,q_ 1}\leq K\{| | u'|^{\alpha} u''|^{1/(\alpha +2)}_{E_{u'}^{\beta_ u},p}| u|_{E,p_ 0}^{1/(\alpha +2)}+| u|_{E,p_ 0}\},\quad \forall u\in C^ 2(E)\quad with\quad u(a)=u(b)=0 \] where: \(E=[a,b]\), \(\beta_ u=| u|_{E,p_ 0}(meas E)^{-1/q_ 1}\), \(E_{u'}^{\beta_ u}=\{x\in E: | u'(x)| >\beta_ u\}\), and moreover: \(p,p_ 0>1\); \(1>p^{-1}+p_ 0^{-1}>\eta >0\); \(\gamma\geq 0\); \(\gamma >\alpha >(p^{-1}+p_ 0^{-1})(1+\eta)-2\); \(q_ 1=pp_ 0(\alpha +2)/(p+p_ 0)\); \(K=2^{1+1/\eta +1/\eta^ 2}[(\gamma +2)/\eta]^{1/\eta^ 2}\).