In this paper we are concerned with a class of optimization problems involving the $p(x)$-Laplacian operator, which arise in imaging and signal analysis. We study the well-posedness of this kind of problems in an amalgam space considering that the variable exponent $p(x)$ is a log-H\"older continuous function. Further, we propose a preconditioned descent algorithm for the numerical solution of the problem, considering a "frozen exponent" approach in a finite dimension space. Finally, we carry on several numerical experiments to show the advantages of our method. Specifically, we study two detailed example whose motivation lies in a possible extension of the proposed technique to image processing.

翻译：本文研究一类出现在成像与信号分析中的、涉及$p(x)$-Laplacian算子的优化问题。我们考虑变指数$p(x)$为对数Hölder连续函数的情形，在融合空间中探讨此类问题的适定性。进一步，我们提出一种用于问题数值求解的预条件下降算法，该算法在有限维空间中采用"冻结指数"方法。最后，我们通过若干数值实验展示所提方法的优势。具体而言，我们详细研究了两个示例，其动机源于将所提技术推广至图像处理领域的可能性。