# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022089
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## Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions

 1 Department of Mathematics, China Jiliang University, Hangzhou 310018, China 2 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author: Kehan Shi

Received  December 2021 Revised  April 2022 Early access May 2022

Fund Project: This work was supported by the Natural Science Foundation of Zhejiang Province, China (LQ21A010010) and the National Natural Science Foundation of China (12001509)

This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed.

Citation: Kehan Shi, Ying Wen. Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022089
##### References:
 [1] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, Journal de Mathématiques Pures et Appliquées, 90 (2008), 201-227. [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, no. 165, American Mathematical Society, 2010. doi: 10.1090/surv/165. [3] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, Journal of Statistical Physics, 95 (1999), 1119-1139.  doi: 10.1023/A:1004514803625. [4] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136.  doi: 10.1007/s002050050037. [5] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, (2001). [6] A. Buades, B. Coll, and J.-M. Morel, A non-local algorithm for image denoising, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), vol. 2, IEEE, 2005, pp. 60–65. [7] C. Carrillo and P. Fife, Spatial effects in discrete generation population models, Journal of Mathematical Biology, 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4. [8] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271-291. [9] C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53-60.  doi: 10.1007/s11856-009-0019-8. [10] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, Journal of Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [11] ————, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137-156. [12] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153–191. [13] A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calculus of Variations and Partial Differential Equations, 59 (2020), 24 pp. doi: 10.1007/s00526-020-01771-z. [14] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-12245-3. [15] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling & Simulation, 6 (2007), 595-630.  doi: 10.1137/060669358. [16] ———, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028. [17] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Modeling & Simulation, 4 (2005), 1091-1115.  doi: 10.1137/050622249. [18] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229. [19] D. O'regan, Fixed-point theory for weakly sequentially continuous mappings, Mathematical and Computer Modelling, 27 (1998), 1-14.  doi: 10.1016/S0895-7177(98)00014-4. [20] P. Radu, D. Toundykov and J. Trageser, A nonlocal biharmonic operator and its connection with the classical analogue, Archive for Rational Mechanics and Analysis, 223 (2017), 845-880.  doi: 10.1007/s00205-016-1047-2. [21] K. Shi, Coupling local and nonlocal diffusion equations for image denoising, Nonlinear Analysis: Real World Applications, 62 (2021), 103362.  doi: 10.1016/j.nonrwa.2021.103362. [22] ————, Image denoising by nonlinear nonlocal diffusion equations, Journal of Computational and Applied Mathematics, 395 (2021), 113605. [23] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace, Indiana University Mathematics Journal, (2006), 1155–1174. doi: 10.1512/iumj.2006.55.2706. [24] J. Sun, J. Li and Q. Liu, Cauchy problem of a nonlocal p-Laplacian evolution equation with nonlocal convection, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 691-702.  doi: 10.1016/j.na.2013.09.023. [25] Y. Wen, L. A. Vese, K. Shi, Z. Guo, and J. Sun, Nonlocal adaptive biharmonic regularizer for image restoration, to appear, Journal of Mathematical Imaging and Vision. [26] Y. Wen, J. Sun and Z. Guo, A new anisotropic fourth-order diffusion equation model based on image feature for image denoising, Inverse Problems & Imaging, (2022). doi: 10.3934/ipi.2022004. [27] Y. Wen and L. A. Vese, Nonlocal adaptive biharmonic regularizer for image restoration, International Symposium on Visual Computing, Springer, 2020, pp. 670–681. doi: 10.1007/978-3-030-64559-5_53. [28] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9 (2000), 1723-1730.  doi: 10.1109/83.869184.

show all references

##### References:
 [1] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, A nonlocal p-Laplacian evolution equation with Neumann boundary conditions, Journal de Mathématiques Pures et Appliquées, 90 (2008), 201-227. [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, no. 165, American Mathematical Society, 2010. doi: 10.1090/surv/165. [3] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, Journal of Statistical Physics, 95 (1999), 1119-1139.  doi: 10.1023/A:1004514803625. [4] P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136.  doi: 10.1007/s002050050037. [5] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, (2001). [6] A. Buades, B. Coll, and J.-M. Morel, A non-local algorithm for image denoising, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), vol. 2, IEEE, 2005, pp. 60–65. [7] C. Carrillo and P. Fife, Spatial effects in discrete generation population models, Journal of Mathematical Biology, 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4. [8] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271-291. [9] C. Cortazar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53-60.  doi: 10.1007/s11856-009-0019-8. [10] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, Journal of Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [11] ————, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137-156. [12] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, (2003), 153–191. [13] A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calculus of Variations and Partial Differential Equations, 59 (2020), 24 pp. doi: 10.1007/s00526-020-01771-z. [14] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-12245-3. [15] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling & Simulation, 6 (2007), 595-630.  doi: 10.1137/060669358. [16] ———, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028. [17] S. Kindermann, S. Osher and P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Modeling & Simulation, 4 (2005), 1091-1115.  doi: 10.1137/050622249. [18] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229. [19] D. O'regan, Fixed-point theory for weakly sequentially continuous mappings, Mathematical and Computer Modelling, 27 (1998), 1-14.  doi: 10.1016/S0895-7177(98)00014-4. [20] P. Radu, D. Toundykov and J. Trageser, A nonlocal biharmonic operator and its connection with the classical analogue, Archive for Rational Mechanics and Analysis, 223 (2017), 845-880.  doi: 10.1007/s00205-016-1047-2. [21] K. Shi, Coupling local and nonlocal diffusion equations for image denoising, Nonlinear Analysis: Real World Applications, 62 (2021), 103362.  doi: 10.1016/j.nonrwa.2021.103362. [22] ————, Image denoising by nonlinear nonlocal diffusion equations, Journal of Computational and Applied Mathematics, 395 (2021), 113605. [23] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace, Indiana University Mathematics Journal, (2006), 1155–1174. doi: 10.1512/iumj.2006.55.2706. [24] J. Sun, J. Li and Q. Liu, Cauchy problem of a nonlocal p-Laplacian evolution equation with nonlocal convection, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 691-702.  doi: 10.1016/j.na.2013.09.023. [25] Y. Wen, L. A. Vese, K. Shi, Z. Guo, and J. Sun, Nonlocal adaptive biharmonic regularizer for image restoration, to appear, Journal of Mathematical Imaging and Vision. [26] Y. Wen, J. Sun and Z. Guo, A new anisotropic fourth-order diffusion equation model based on image feature for image denoising, Inverse Problems & Imaging, (2022). doi: 10.3934/ipi.2022004. [27] Y. Wen and L. A. Vese, Nonlocal adaptive biharmonic regularizer for image restoration, International Symposium on Visual Computing, Springer, 2020, pp. 670–681. doi: 10.1007/978-3-030-64559-5_53. [28] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9 (2000), 1723-1730.  doi: 10.1109/83.869184.
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