Elliptic Partial Differential Equations

By M. Marin, A. Öchsner. Numerical Methods for Engineers and Scientists, 2018

2,365 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1b1d31f1614194b7f21679f40248b4532f09cd4b

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Citations in the corpus

Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation (2021, neutral): “Let us also recall the Gagliardo-Nirenberg inequality [27], if N ≥ 2 and p ∈ [2, 2N N−2 ) then ‖f ‖Lp(RN ) ≤ C(N,p)‖∇f ‖ β L2(RN ) ‖f ‖ L2(RN ) , with β =N (”
Where Sobolev interacts with Gagliardo–Nirenberg (2019, methodology): “1) was established by Gagliardo [8] and Nirenberg [12].”
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis ✩ (2016, background): “The following Gagliardo–Nirenberg interpolation inequality also plays a key role in our proof (see [24,37] for detail).”
Regularity Theory for Elliptic PDE (2022, methodology): “Moreover, we believe that the present selection of results and techniques complements nicely other books on elliptic PDE such as [Eva98], [HL97], and [Kry96], as well as the very recent book [ACM18].”
Time fractional diffusion equations: solution concepts, regularity, and long-time behavior (2019, background): “in the case α = 1) there is a powerful theory of a priori estimates, often referred to as De Giorgi-Nash-Moser theory, which provides local and global estimates for weak solutions of the respective equations such as local and global boundedness, Harnack and weak Harnack inequalities as well as Hölde”
Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier–Stokes Equations with Vacuum (2013, methodology): “Next, the following well-known Gagliardo-Nirenberg inequality (see [26]) will be used later.”
Dissipative Gradient Nonlinearities Prevent δ$\delta$ ‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models (2024, positive): “In order to prove relation (40), we use the Gagliardo–Nirenberg inequality (see [38]) in conjunction with (16a) and (37), obtaining where in the second-last step we employed relation (11), and in the last one Young’s inequality, justified by relation (16b).”
Embeddings for A-weakly differentiable functions on domains (2017, background): “It was much later that Gagliardo [24] and Nirenberg [36] independently showed with new methods that (1.”
Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks (2018, background): “Nirenberg ([32]), but it is not developed in detail.”
Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation (2018, background): “[24]) then allows us to improve the result of Proposition 4.”
Multiple normalized solutions for a Sobolev critical Schrödinger equation (2020, neutral): “for all f ∈H , see [33].”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, neutral): “[46] or [72]): for every k ∈ N, there exists C = C(k, d,Λ) <∞ such that, for every R > 0 and a-harmonic function u on BR, ∥∇ku∥ L∞(BR/2) ⩽ C Rk ∥u∥L2(BR) .”
Gagliardo–Nirenberg inequalities and non-inequalities: The full story (2017, background): “Nirenberg [10] established the interpolation inequality ‖ f ‖Wk,p .”