The method in the works mentioned above, local in nature, seems to use in a essential way the nondegeneracy of the critical points. A precise concentration statement around a global minimum, possibly degenerate for this solution is established in . A question raised to us by Changfeng Gui, which motivates the present work, is whether one can find solutions which concentrate around local minima not necessarily nondegenerate. As we will see, the answer is affirmative, and moreover the same is true if one considers the elliptic problem in an arbitrary domain in RN with zero boundary conditions.
Our main result for equation 0. Theorem 0. In particular, V is not required to be bounded or to belong to a Kato class.
The energy functional associated to equation 0. Indeed, a global assumption like 0. What we do in our situation is to build a convenient modification of the energy, in such a way that the functional satisfies P. To ilustrate this point, we will see in the last section how local properties of an unbounded domain lead to existence and localization of solutions to elliptic problems. As an example of the situation covered by Theorem 3. In the framework of Theorem 0.
Associated to equation 0. We will define a modification of this functional which satisfies the P. Let us notice that the function f considered in the intro- duction satisfies the properties given for g, except for g3 ii. This last assumption implies that J satisfies the Palais Smale condition as we show next. Lemma 1.
Then un has a convergent subsequence. This convergence is actually strong. Condition g4 implies that the critical value c can be characterized in a simpler way, as has been essentially established in  and . We provide a proof for completeness. In view of g1 and 1. A proof of this fact was given in , so that we only provide a sketch.
This will provide a path with values strictly less that c1. As we already mentioned, in order to localize the mountain passes we had modified the function f.
We consider the functional I defined in 1. Lemma 2.
Crucial step in the proof of this fact is the following Proposition 2. Proof of Theorem 0. By Proposition 2. To obtain 0. From 2. See the works of Gidas, Ni and Nirenberg  and . It follows from 2. Since w0 decays exponentially, relation 0. Again from 2.
Symmetry And Monotonicity Properties For Positive Solutions Of Semi-Linear Elliptic PDE'S
Then v is a critical point of J. In particular, J v Proof. Using this in 2. But this contradices the previous lemma, and the proof of the 2.
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Hence I v claim, i. Then, it turns out, that after passing to a subsequence vn converges in the C 2 sense over compacts to a solution v in H 1 RN of equation 2. The function v has a local maximum at zero, it is radially symmetric and radially decreasing, as the arguments in  show. Then 0 is a nondegenerate global maximum. Using this fact, a slight variation of the argument in the proof of Lemma 2. Manzoni, and A. We present different modeling approaches involving the Navier—Stokes equations to represent the incompressible fluid and Darcy of Fochheimer equations in the porous region.
We discuss and compare different coupling strategies based on the so—called penalization method or on the application of the transmission conditions of Beavers—Joseph—Saffman. After giving some results on the well—posedness of the resulting coupled models, we illustrate a domain decomposition framework to set up possible iterative methods to compute their finite element solution.
We discuss the effectiveness and robustness of such algorithms showing several numerical results.
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Finally, we present some possible applications of the models that we have studied. Soft materials such as polymers and biological tissues have several engineering and biomechanical applications. These materials exhibit complex mechanical characteristics and the need to accurately predict their behavior has posed a tremendous challenge for scientists and engineers. Less than a half century old, continuum computational modeling of soft materials is undoubtedly still in its infancy and its projected outlook remains promising for improving humanity's quality of life.
Continued advances in computers and computational methods are increasing our ability to handle large amounts of data and to model complex phenomena. Modeling of soft materials has a vital role to play in the development of the needed mathematical models and analyses. Because of the incredible complexity of the biochemophysical aspects of soft tissues and the chemophysical aspects of polymers, this type of modeling requires increased interdisciplinary and multidisciplinary research that brings scientists from various fields together in teams, both in research and education.
Publications of Tobias Weth
In this talk, I shall present a seamless, fully variational constitutive model capable of capturing several complex mechanical characteristics exhibited by such materials. Future directions of this work may lead to the formulation of head-injury criteria for medical, governmental, and industrial applications; addressing the definition of clinical-biomechanical injury thresholds and tolerances; the simulation of a wide range of injuries, including blast-induced TBI and the effects of growing tumors; neurosurgical simulations; and the design and the assessment of effective protective devices, such as helmets including honeycomb materials, polymers, or foam padding.
In the first part i will present what it has been done in the subcritical case. The second part will concern the supercritical case. We shall recall some recent concentration-compactness and quantization results for elliptic equations having critical nonlinearity, which arise in conformal geometry and functional analysis.
By looking at a model case, we shall try to discuss how these results can be used to prove existence for some important equations mean field equation, prescribed Q—curvature equation. We shall also see several open problems. Swimming microorganisms often exhibit cooperative motion of active elements attached on the cell membrane, such as the metachronal waves of cilia and the rotation of bundled flagella. In order to understand the role of long-range hydrodynamic interactions in their collective dynamics, we introduce a simple and generic model of active microfluidic rotors arrayed on a substrate.
The model shows a variety of dynamical patterns such as global synchronization, frustrated disorder, and turbulent spiral waves. We discuss the results in connection with recent experimental observations on bacterial carpets, as well as with theoretical developments on non-locally coupled oscillators. Golestanian, arXiv I will discuss symmetry properties of a certain class of solutions to semilinear elliptic equations. These solutions are characterized variationally by bounds on their Morse index.
In the case where the underlying domain is radial, the solutions do not always inherit the full radial symmetry of the domain but at least axial symmetry. In some cases, the symmetry properties lead, a posteriori, to nonexistence results.