<p>We study an optimal control problem for a nonlinear elliptic anisotropic <i>p</i>-Laplace equation with control constraints and Neumann boundary conditions. The matrix-valued coefficients <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1044_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">$A_{sym}\in L^\infty (\Omega ;\mathbb {S}_{sym}^N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mrow> <mi mathvariant="italic">sym</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <msubsup> <mi mathvariant="double-struck">S</mi> <mrow> <mi mathvariant="italic">sym</mi> </mrow> <mi>N</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> we take as controls and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1044_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="147" /> </InlineMediaObject> <EquationSource Format="TEX">$A_{skew}\in L^q(\Omega ;\mathbb {S}^N_{skew})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mrow> <mi mathvariant="italic">skew</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <msubsup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi mathvariant="italic">skew</mi> </mrow> <mi>N</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We show that, in spite of unboundedness of the differential operator, the considered Neumann problem admits at least one weak solution and the corresponding OCP is well-possed and solvable.</p>

On optimal controls in coefficients for ill-posed elliptic Neumann boundary value problem with anisotropic p-Laplace operator

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发表日期：

2025年4月2日

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Peter Kogut,
Olha Kupenko

摘要

We study an optimal control problem for a nonlinear elliptic anisotropic p-Laplace equation with control constraints and Neumann boundary conditions. The matrix-valued coefficients $A_{sym}\in L^\infty (\Omega ;\mathbb {S}_{sym}^N)$ $A_{sym} \in L^{\infty} (Ω; S_{sym}^{N})$ we take as controls and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix $A_{skew}\in L^q(\Omega ;\mathbb {S}^N_{skew})$ $A_{skew} \in L^{q} (Ω; S_{skew}^{N})$ . We show that, in spite of unboundedness of the differential operator, the considered Neumann problem admits at least one weak solution and the corresponding OCP is well-possed and solvable.