No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

<div class="abstract"> <p> <div>&lt;div class="abstract"&gt; &lt;div&gt;&lt;p&gt;This paper deals with second-order optimality conditions for a quasilinear elliptic&lt;br /&gt;control problem with a nonlinear coefficient in the principal part that is countably PC&lt;sup&gt;2&lt;/sup&gt;&lt;br /&gt;(continuous and C&lt;sup&gt;2&lt;/sup&gt; apart from countably many points). We prove that the control-to-state&lt;br /&gt;operator is continuously differentiable even though the nonlinear coefficient is non-smooth.&lt;br /&gt;This enables us to establish &amp;ldquo;no-gap&amp;rdquo; second-order necessary and sufficient optimality&lt;br /&gt;conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition&lt;br /&gt;only differs from the necessary one in the fact that the inequality is strict. A condition that&lt;br /&gt;is equivalent to the second-order sucient optimality condition and could be useful for&lt;br /&gt;error estimates in, e.g., finite element discretizations is also provided.&lt;/p&gt; &lt;/div&gt; &lt;/div&gt;</div> </p> </div>